
Class 
Book 

COPYRIGHT DEPOSm 



/ 



THE THEORY 



OF 



ENGINEEEING DRAWING 



BY 

ALPHONSE A. ADLER, B.S., M.E. 

Member American Society of Mechanical Engineers; Assistant Professor of 
Mechanical Drawing and Design, Polytechnic Institute, Brooklyn, N. Y. 



273 ILLUSTRATIONS 



SECOND EDITION, CORRECTED 




NEW YORK 

D. VAN NOSTRAND COMPANY 

25 PARK PLACE 

1915 



^Vx 



\<^- 



Copyright, 1912, by 
D. VAN NOSTRAND COMPANY 



Copyright, 1915, by 
D. VAN NOSTRAND COMPANY 



THE SCIENTIFIC PRESS 

HOBERT DRUMMOND AND COMPANY 

BROOKLYN. N. Y. 

JUL I iyi5 

©CIA406530 



X. 



PREFACE TO THE FIRST EDITION 



Although the subject matter of this volume is, in large 
measure, identical with that of many treatises on descriptive 
geometry, the author has called it ''Theory of Engineering 
Drawing," beheving that this title indicates better than could 
any other, the ultimate purpose of the book. That texts on 
descriptive geometry appear with some degree of frequency, 
with but few, if any, additions to the theory, indicates that 
teachers are aware of certain weaknesses in existmg methods of 
presenting the subject. It is precisely these weaknesses that the 
present work aims to correct. 

The author emphasizes the fact that the student is concerned 
with the representation on a plane of objects in space of three 
dimensions. The analysis, important as it is, has for its primary 
purpose the development of methods for such representation 
and the interpretation of the resulting drawings. It is nowhere 
regarded as an end in itself. The number of fundamental prin- 
ciples has been reduced to a minimum; indeed it will be found 
that the entire text is based on the problem of finding the piercing 
point of a given line on a given surface, and a few additional 
operations. The accepted method of presenting the subject, 
IS to start with a set of definitions,to consider in detail the ortho- 
graphic projection of a point, and then, on the foundation thus 
laid, to build the theory of the projection of lines, surfaces, and 
sohds. Logical and beautiful as this systematic development 
may be, it nevertheless presents certain inherent difficulties, chief 
of which is that the student is confronted at the outset with that 
most abstract of all abstractions, the mathematical point. In 
this volume the order of presentation is reversed and th(^ reader 
IS asked to consider first some concrete object, a box, for instance, 
the study of which furnishes material of use in the later discussion 
of its bounding surfaces and lines. 

The '' Theory of Engineering Drawing " is divided into four 



IV PREFACE 

parts. Part I treats of oblique projection, orthographic projec- 
tion, and a special case of the latter, axonometric projection. The 
student is advised to give special attention to the classification 
at the end of this section, because it gives a complete outline of 
the entire subject. Part II contains a variety of problems of 
such nature as to be easily understood by those whose training 
has not extended to the more highly speciahzed branches of com- 
mercial or engineering practice. Part III considers convergent 
projective line drawing, more familiar under the name of perspec- 
tive. Part IV has to do \vith the pictorial effects of illumination, 
since a knowledge of shades and shadows is frequently required 
in the preparation of complicated drawings. 

No claim is made to originality of subject matter, but it is 
not possible to acknowledge indebtedness to individual writers, 
for the topics discussed have been widely studied, and an historical 
review is here out of place. The author wishes, however, to express 
his sense of obligation to Professor WiUiam J. Berry of the Depart- 
ment of Mathematics in the Polytechnic Institute of Brooklyn 
for his criticism of Chapters IX and X, and other assistance, 
and to Professor Ernest J. Streubel of the Department of English 
for his untiring efforts in preparing the manuscript for the press. 

A. A. A. 

Polytechnic Institute of Brooklyn, 
October, 1912. 



PREFACE TO THE SECOND EDITION 



In this edition the author has limited the revision to the cor- 
rection of minor errors. These have been suggested by various 
readers, to whom thanks are hereby given. 

A. A. A. 
Polytechnic Institute of Brooklyn, 
February, 1915. 



CONTENTS 



PART I 

THE PRINCIPLES OF PARALLEL PROJECTING-LINE 

DRAWING 

CHAPTER I 

INTRODUCTORY 

AKT. PAGE 

101. Nature of drawing 3 

102. Science and art of drawing 4 

103. Magnitude of objects 4 

104. Commercial application of drawing 4 

CHAPTER II 
OBLIQUE PROJECTION 

201. Nature of obKque projection 6 

202. Oblique projection of lines parallel to the plane of projection 7 

203. Oblique projection considered as a shadow 8 

204. Obhque projection of Hues perpendicular to the plane of projection . 9 

205. Obhque projection of the combination of parallel and perpendicular 

lines to the plane of projection 10 

206. Obhque projection of circles 11 

207. Obhque projection of inclined lines and angles 12 

208. Representation of visible and invisible lines 13 

209. Drawings to scale 14 

210. Examples of oblique projection 14 

211. Distortion of obhque projection 20 

212. Commercial application of obhque projection 21 

CHAPTER III 
ORTHOGRAPHIC PROJECTION 



301. Nature of orthographic projection 2.") 

302. Theory of orthographic j)rojection 20 

303. Revolution of the horizontal phuie 27 

V 



vi . CONTENTS 

ART. PAGE 

304. Position of the eye 27 

305. Relation of size of object to size of projection 28 

306. Location of object with respect to the planes of projection 28 

307. Location of projections with respect to each other 29 

308. Dimensions on a projection 29 

309. Comparison between oblique and orthographic projection 29 

310. Orthographic projection considered as a shadow 30 

311. Profile plane 30 

312. Location of profiles 31 

313. Section plane 33 

314. Supplementary plane 34 

315. Angles of projection 36 

316. Location of observer in constructing projections 36 

317. Application of angles of projections to drawing 37 

318. Commercial application of orthographic projection 38 

CHAPTER IV 
AXONOMETRIC PROJECTION 

401. Nature of isometric projections 45 

402. Theory of isometric projection 46 

403. Isometric projection and isometric drawing 47 

404. Direction of axes 48 

405. Isometric projection of circles 48 

406. Isometric projection of incHned fines and angles 49 

407. Isometric graduation of a circle 49 

408. Examples of isometric drawing 51 

409. Dimetric projection and dimetric drawing 54 

410. Trimetric projection and trimetric drawing 55 

411. Axonometric projection and axonometric drawing 56 

412. Commercial application of axonometric projection 56 

413. Classification of projections 57 



PART II 



GEOMETRICAL PROBLEMS IN ORTHOGRAPHIC 
PROJECTION 

CHAPTER V 

REPRESENTATION OF LINES AND POINTS 

501. Introductory 61 

502. Representation of the fine 62 

503. Line fixed in space by its projections 64 

504. Orthographic representation of a fine 65 

505. Transfer of diagrams from orthographic to obUque projection, 66 



CONTENTS vii 

ART. PAGE 

506. Piercing points of lines on the principal planes. . > 66 

507. Nomenclature of projections 68 

508. Representation of points 68 

509. Points lying in the principal planes 69 

510. Mechanical representation of the principal planes 69 

511. Lines parallel to the planes of projection 70 

512. Lines lying in the planes of projection 71 

513. Lines perpendicular to the planes of projection 72 

514. Lines in all angles 72 

515. Lines with coincident projections 74 

516. Points in all angles 75 

517. Points with coincident projections 75 

518. Lines in profile planes 75 

CHAPTER VI 
REPRESENTATION OF PLANES 

601. Traces of planes parallel to the principal planes 80 

602. Traces of planes parallel to the ground line 80 

603. Traces of planes perpendicular to one of the principal planes 82 

604. Traces of planes perpendicular to both principal planes 83 

605. Traces of planes inclined to both principal planes 83 

606. Traces of planes intersecting the ground Hne 84 

607. Plane fixed in space by its traces 84 

608. Transfer of diagrams from orthographic to obHque projection 84 

609. Traces of planes in all angles 86 

610. Projecting plane of fines 86 

CHAPTER VII 
ELEMENTARY CONSIDERATIONS OF LINES AND PLANES 

701. Projection of lines parallel in space 89 

702. Projection of lines intersecting in space 89 

703. Projection of lines not intersecting in space 90 

704. Projection of lines in obfique planes 91 

705. Projection of lines parallel to the principal planes and lying in an 

oblique plane 92 

706. Projection of lines perpendicular to given planes 95 

707. Revolution of a point about a fine 96 

CHAPTER VIII 

PROBLEMS INVOLVING THE POINT, THE LINE, AND THE 

PLANE 

801. Introductory 99 

802. Solution of problems 99 

803. Problem 1. To draw a fine through a given point paraUel to a given 

lino 100 



viii CONTENTS 

ART. PAGE 

804, Problem 2. To draw a line intersecting a given line at a given point . . 100 

S05. Problem 3. To find where a given line pierces the principal 

planes 101 

806. Problem 4. To pass an obhque plane through a given obhque hne. . 102 

807. Special cases of the preceding problem 102 

808. Problem 5. To i)ass an obhque plane through a given point 103 

809. Problem 6. To find the intersection of two planes, oblique to each 

other and to the principal planes 104 

810. Special case of the preceding problem 104 

811. Problem 7. To find the corresponding projection of a given point 

lying in a given obhque plane, when one of its projections is 
given 104 

812. Special case of the preceding problem 105 

813. Problem 8. To draw a plane which contains a given point and is 

paraUel to a given plane 106 

814. Problem 9. To draw a line perpendicular to a given plane through 

a given point 107 

815. Special case of the precechng problem 108 

816. Problem 10. To draw a plane through a given point perpendicular to 

a given line 108 

817. Problem 11. To pass a plane through three given points not in the 

same straight line 109 

818. Problem 12. To revolve a given point, not in the principal planes, 

about a line h'ing in one of the principal planes 110 

819. Problem 13. To find the true distance between two points in space 

as given by their projections. First method. Case 1 Ill 

820. Case 2 \ 112 

821. Problem 13. To find the true distance between two points in space 

as given by their projections. Second method. Case 1 . . . 113 

822. Case 2 ". 113 

823. Problem 14. To find where a given Hne pierces a given plane 114 

824. Problem 15. To find the distance of a given point from a given 

plane 115 

825. Problem 16. To find the distance from a given point to a given hne.... 115 

826. Problem 17. To find the angle between two given intersecting 

hues 116 

827. Problem 18. To find the angle between two given planes 117 

828. Problem 19. To find the angle between a given plane and one of 

the principal planes 118 

829. Problem 20. To draw a plane parallel to a given plane at a given 

distance from it 119 

830. Problem 21. To project a given hne on a given plane 120 

831. Problem 22. To find the angle between a given hne and a given 

plane 121 

832. Problem 23. To find the shortest distance between a pair of skew 

lines 122 

833. Apphcation to other problems 125 



CONTENTS IX 

ART. PAGE 

834. Problem 24. Through a given point, draw a Hne of a given length, 

making given angles with the planes of projection 125 

835. Problem 25. Through a given point, draw a plane, making given 

angles with the principal planes 127 

836. Problem 26. Through a given hne, in a given plane, draw another 

hne, intersecting it at a given angle 129 

837. Problem 27. Through a given hne, in a given plane, pass another 

plane making a given angle with the given plane 130 

838. Problem 28. To construct the projections of a circle lying in a given 

obhque plane, of a given diameter, its centre in the plane being 
known 131 



CHAPTER IX 
CLASS_ FICATION OF LINES 

901. Introductory 144 

902. Straight hne 144 

903. Singly curved hne , .' 144 

904. Representation of straight and singly curved lines 144 

905. Cu-cle 145 

906. EUipse 146 

907. Parabola 147 

908. Hyperbola 147 

909. Cycloid 148 

910. Epicycloid 149 

911. Hypocycloid 150 

912. Spiral 151 

913. Doubly curved line 151 

914. Representation of doubly curved lines 151 

915. Hehx 152 

916. Classification of hnes 154 

917. Tangent 154 

918. Construction of a tangent 154 

919. To find the point of tangency 155 

920. Direction of a curve 156 

921. Angle between curves 156 

922. Intersection of lines 156 

923. Order of contact of tangents 157 

924. Osculating circle 158 

925. Osculating i)lane 158 

926. Point of inflexion. Inflexional tangent 159 

927. Normal 159 

928. Rectification . . 159 

929. Involute and Evolute . . 160 

930. Involute of the circle H)l 



X CONTENTS 

CHAPTER X 
CLASSIFICATION OF SURFACES 

ART. PAGE 

1001. Introductory 165 

1002. Plane surface 165 

1003. Conical surface 166 

1004. Cone 166 

1005. Representation of the cone 167 

1006. To assume an element on the surface of a cone 168 

1007. To assume a point on the surface of a cone 168 

1008. Cylindrical surface 169 

1009. Cyhnder .169 

1010. Representation of the cyhnder 170 

1011. To assume an element on the surface of a cyhnder 170 

1012. To assume a point on the surface of a cyhnder 171 

1013. Convolute surface 171 

1014. Obhque helicoidal screw surface 173 

1015. Right hehcoidal screw surface 174 

1016. Warped surface 174 

1017. Tangent plane ". 175 

1018. Normal plane 175 

1019. Singly curved surface 176 

1020. Doubly curved surface 176 

1021. Singly curved surface of revolution 176 

1022. Doubly curved surface of revolution 176 

1023. Revolution of a skew hne 177 

1024. Meridian plane and meridian hne 177 

1025. Surfaces of revolution having a common axis 177 

1026. Representation of the doubly curved surface of revolution 178 

1027. To assume a point on a doubly curved surface of revolution 178 

1028. Developable surface 179 

1029. Ruled surface 179 

1030. Asymptotic surface 179 

1031. Classification of surfaces 180 

CHAPTER XI 

INTERSECTIONS OF SURFACES BY PLANES, AND THEIR 

DEVELOPMENT 

1101. Introductory 184 

1102. Lines of intersection of sohds by planes 185 

1103. Development of surfaces 185 

1104. Developable surfaces 185 

1105. Problem 1. To find the line of intersection of the surf aces of a 

right octagonal prism with a plane inclined to its axis 186 



CONTENTS xi 

ART. PAGE 

1106. Problem 2. To find the developed surfaces in the preceding prob- 

lem 187 

1107. Problem 3. To find the fine of intersection of the surface of a right 

circular cyUnder with a plane inclined to its axis 188 

1108. Problem 4. To find the developed surf ace in the preceding problem . 189 

1109. AppHcation of cyhndrical surfaces 189 

1110. Problem 5. To find the line of intersection of the sm-faces of a 

right octagonal pyramid with a plane inclined to its axis. . . 190 

1111. Problem 6. To find the developed surfaces in the preceding problem 191 

1112. Problem 7. To find the fine of intersection of the surface of a right 

circular cone with a plane inclined to its axis 192 

1113. Problem 8. To find the developed surface in the preceding problem . 193 

1114. AppHcation of conical surfaces 194 

1115. Problem 9. To find the line of intersection of a doubly curved 

surface of revolution with a plane inchned to its axis 194 

1116. Problem 10. To find the fine of intersection of a bell-siu-face with a 

plane 195 

1117. Development by triangulation 196 

1118. Problem 11. To develop the surfaces of an obHque hexagonal 

pyramid 196 

1119. Problem 12. To develop the surface of an oblique cone 197 

1120. Problem 13. To develop the surface of an oblique cyHnder 198 

1121. Transition pieces 199 

1122. Problem 14. To develop the sm-face of a transition piece connect- 

ing a circular opening with a square opening 200 

1123. Problem 15. To develop the sm-face of a transition piece connect- 

ing two eUiptical openings whose major axes are at right angles 
to each other 201 

1124. Development of doubly curved sm-faces by approximation 202 

1125. Problem 16. To develop the surface of a sphere by the gore 

method 203 

1126. Problem 17. To develop the surface of a sphere by the zone 

method 204 

1127. Problem 18. To develop a doubly curved surface of revolution by 

the gore method 205 



CHAPTER XII 

INTERSECTIONS OF SURFACES WITH EACH OTHER AND 
THEIR DEVELOPMENT 

1201. Introductory 210 

1202. Problem 1. To find the line of intersection of the surfaces of two 

prisms 211 

1203. Problem 2. To find the developments in the i)receding prohhMn. .211 

1204. Problem 3. To find the line of intersection of two cylindrical sur- 

faces of revolution whose axes intersect at a right angle .... 212 



xii CONTENTS 

•ART. PAGE 

1205. Problem 4. To find the developments in the preceding problem . . . . 213 

1206. Problem 5. To find the line of intersection of two cyUndrical 

surfaces of revolution whose axes intersect at any angle. . 213 

1207. Problem 6. To find the developments in the preceding problem. . . 214 

1208. AppUcation of intersecting cylindrical surfaces to pipes 214 

1209. Problem 7. To find the line of intersection of two cyHndrical sur- 

faces whose axes do not intersect 215 

1210. Problem 8. To find the developments in the preceding problem. . . 215 

1211. Intersection of conical surfaces 216 

1212. Problem 9. To find the line of intersection of the surfaces of two 

cones whose bases may be made to he in the same plane, and 
whose altitudes differ 216 

1213. Problem 10. To find the hne of intersection of the surfaces of two 

cones whose bases may be made to lie in the same plane, and 
whose altitudes are equal 218 

1214. Problem 11. To find the line of intersection of the surfaces of two 

cones whose bases he in different planes 219 

1215. Types of lines of intersection for surfaces of cones 221 

1216. Problem 12. To find the hne of intersection of the surfaces of a 

cone and a cylinder of revolution when their axes intersect at 

a right angle 222 

1217. Problem 13. To find the line of intersection of the surfaces of a cone 

and a cylinder of revolution when their axes intersect at any 
angle 222 

1218. Problem 14. To find the hne of intersection of the surfaces of an 

obhque cone and a right cylinder 223 

1219. Problem 15. To find the developments in the preceding problem . 224 

1220. Problem 16. To find the line of intersection of the surfaces of an 

obhque cone and a sphere 224 

1221. Problem 17. To find the line of intersection of the surfaces of a 

cyhnder and a sphere 225 

1222. Problem 18. To find the line of intersection of two doubly curved 

surfaces of revolution whose axes intersect 226 

1223. Commercial application of methods 226 



CONTENTS xiii 



PART III 



THE PRINCIPLES OF CONVERGENT PROJECT IN G-LINE 

DRAWING 

CHAPTER XIII 

PERSPECTIVE PROJECTION 

ART. PAGE 

1301. Introductory , 235 

1302. Scenographic projection 235 

1303. Linear perspective 236 

1304. Visual rays and visual angle 236 

1305. Vanishing point 237 

1306. Theory of perspective projection 237 

1307. Aerial perspective 237 

1308. Location of picture plane 237 

1309. Perspective of a hne 238 

1310. Perspectives of lines perpendicular to the horizontal plane 239 

1311. Perspectives of Hnes parallel to both principal planes 239 

1312. Perspectives of lines perpendicular to the picture plane 240 

1313. Perspectives of parallel lines, inclined to the picture plane 240 

1314. Horizon 241 

1315. Perspective of a point 242 

1316. Indefinite perspective of a Une 242 

1317. Problem 1. To find the perspective of a cube by means of the 

piercing points of the visual rays on the picture plane 244 

1318. Perspectives of intersecting lines 245 

1319. Perpendicular and diagonal 245 

1320. To find the perspective of a point by the method of perpendiculars 

and diagonals 246 

1321. To find the perspective of a line by the method of perpendiculars 

and diagonals 248 

1322. Revolution of the horizontal plane 249 

1323. To find the perspective of a point when the horizontal plane is 

revolved 249 

1324. To find the perspective of a line when the horizontal phxne is 

revolved 250 

1325. Location of diagonal vanishing points 251 

1326. Problem 2. To find the j)orsi)ective of a cube by the method of 

perpendiculars and diagonals 251 

1327. Problem 3. To find the perspective of a hexagonal prism 253 

1328. Problem 4. To find the perspective of a pyramid superimposed on 

a square base 254 

1329. Problem 5. To find the perspective of an arch 254 

1330. Problem 6. To find the perspective of a building 256 

1331. Comnuircial ajjphcation of perspective 25S 

1332." Classification of projections 260 



xiv CONTENTS 

PART IV 

PICTORIAL EFFECTS OF ILLUMINATION 

CHAPTER XIV 

PICTORIAL EFFECTS OF ILLUMINATION IN ORTHO- 
GRAPHIC PROJECTION 

ART. PAGE 

140L Introductory 265 

1402. Line shading applied to straight lines 265 

1403. Line shading apphed to curved lines 266 

1404. Line shading applied to sections 267 

1405. Line shading appHed to convex surfaces 267 

1406. Line shading apphed to concave surfaces 268 

1407. Line shading apphed to plane surfaces 268 

1408. Physiological effect of hght 268 

1409. Conventional direction of light rays 269 

1410. Shade and shadow 269 

1411. Umbra and penumbra 269 

1412. Apphcation of the physical principles of Ught to drawing 270 

1413. Shadows of hues 270 

1414. Problem 1. To find the shadow cast by a cube which rests on a 

plane 271 

1415. Problem 2. To find the shadow cast by a pjTamid, in the principal 

planes 272 

1416. Problem 3. To find the shade and shadow cast by an octagonal 

prism having a superimposed octagonal cap 273 

1417. Problem 4. To find the shade and shadow cast by a superimposed 

circular cap on a cyhnder 275 

1418. High-hght 275 

1419. Incident and reflected rays 276 

1420. Problem 5. To find the high light on a sphere 276 

1421. Multiple high hghts 277 

1422. High hghts on cyhndrical or conical surfaces 277 

1423. Aerial effect of illumination 277 

1424. Graduation of shade 278 

1425. Shading rules 278 

1426. Examples of graduated shades 279 

CHAPTER XV 

PICTORIAL EFFECTS OF ILLUMINATION IN PERSPECTIVE 
PROJECTION 

1501. Introductory 282 

1502. Problem 1. To draw the perspective of a rectangular prism and its 

shadow on the horizontal plane 282 

1503. General method of finding the perspective of a shadow 284 



CONTENTS XV 

ART. PAGE 

1504. Perspectives of parallel rays of light 284 

1505. Perspective of the intersection of the visual plane on the plane 

receiving the shadow 285 

1506. Apphcation of the general method of finding the perspective of a 

shadow 285 

1507. Problem 2. To draw the perspective of an obeUsk with its shade 

and shadow 286 

1508. Commercial application of the pictorial effects of illumination in 

perspective 287 



PART I. 



PART I 

PRINCIPLES OF PARALLEL PROJECTING-LINE 

DRAWING 



CHAPTER I 

INTRODUCTORY 

101. Nature of drawing. Drawing has for its purpose the 
exact graphic representation of objects in space. The first 
essential is to have an idea, and then a desire to express it. Ideas 
may be expressed in words, in pictures, or in a combination of 
both words and pictures. If words alone are sufficient to express 
the idea, then language becomes the vehicle of its transmission. 
When the idea relates to some material object, however, a drawing 
alone, without additional information may satisfy its accurate 
conveyance. Further, some special cases require for their expres- 
sion a combination of both language and drawing. 

Consider, for purposes of illustration, a maple block, 2 inches 
thick, 4 inches wide and 12 inches long. It is easy to conceive this 
block of wood, and the mere statement, alone, specifies the object 
more or less completely. On the other hand, the modern news- 
paper printing press can not be completely described by language 
alone. Anyone who has ever seen such a press in operation, 
would soon realize that the intricate mechanism could not be 
described in words, so as to make it intelligible to another without 
the use of a drawing. Even if a drawing is employed in tliis 
latter case, the desired idea may not be adequately presented, 
since a circular shaft is drawn in exactly the same way, whether 
it be made of wood, brass, or steel. Appended notes, in such 
cases, inform the constructor of the mat(Mial to use. From 
the foregoing, it is evident, that drawing cannot l)ecome a uni- 
versal language in engineering, unless the ap])(Mided descriptions 
and specifications have the same meaning to all. 



4 PARALLEL PROJECTING-LINE DRAWING 

102. Science and art of drawing. Drawing is both a science 
and an art. The science affects such matters as the proper 
arrangement of views and the manner of their presentation. 

Those who are famihar Tvdth the mode of representation used, 
will obtain the idea the maker desired to express. It is a science, 
because the facts can be assimilated, classified, and presented 
in a more or less logical order. In this book, the science of 
drawing will engage most of the attention; only such of the 
artistic side is included as adds to the ease of the interpretation 
of the drawing. 

The art lies in the skilful application of the scientific prin- 
ciples involved to a definite purpose. It embraces such topics 
as the thickness or weight of lines, whether the outline alone is 
to be drawn, or whether the object is to be colored and shaded 
so as to give it the same appearance that it has in nature. 

103. Magnitude of objects. Objects visible to the eye are, 
of necessity, solids, and therefore require the three principal 
dimensions to indicate their magnitude — length, breadth, and 
thickness. If the observer places himself in the proper position 
while viewing an object before him, the object impresses itself 
on him as a whole, and a mental estimate is made from the one 
position of the observer as to its form and magnitude. Naturally, 
the first task will be to represent an object in a single view, show- 
ing it in three dimensions, as a solid. 

104. Commercial application of drawing. It must be 
remembered that the function of drawing is graphically to 
present an idea on a flat surface — like a sheet of paper for 
instance — so as to take the place of the object in space. The 
reader's imagination supphes such deficiency as is caused by the 
absence of the actual object. It is, therefore, necessary to 
study the various underlying principles of drawing, and, then, 
apply them as daily experience dictates to be the most direct 
and accurate way of their presentation. In any case, only one 
interpretation of a drawing should be possible, and if there is 
a possibility of ambiguity arising, then a note should be made 
on the drawing calling attention to the desired interpretation. 



INTRODUCTORY 



QUESTIONS ON CHAPTER I 

1. In what ways ma}^ ideas be transmitted to others? 

2. What topics are embraced in the science of drawing? 

3. What topics are included in the art of drawing? 

4. How many principal dimensions are required to express the magni- 

tude of objects? What are they? 

5. What is the function of drawing? 

6. Is the reader's imagination called upon when interpreting a drawing? 

Why? 



CHAPTER II 

OBLIQUE PROJECTION 

201. Nature of oblique projection. Suppose it is desired 
to draw a box, 6" wide, 12'' long and 4" high, made of wood 
f thick. Fig. 1 shows this dra^\TL in obUque projection. The 
method of making the drawing wdll first be shown and then 
the theory on w^hich it is based will be developed. A rectangle 
abed, 4:''XQ", is laid out, the 6'' side being horizontal and the 
4:" side being vertical. From three corners of the rectangle, 




Fig. 1. 



lines ae, bf, and eg are drawn, making, in this case, an angle of 
30° with the horizontal. The length 12" is laid off on an 
inclined line, as eg. The extreme Umiting lines of the box are 
then fixed by the addition of two lines ef (horizontal) and fg 
(vertical). The thickness of the wood is represented, and the 
dimensions showing that it is J'' thick indicate the direction 
in which they are laid off. The reason for the presence 
of such other additional lines, is that they show the actual 
vconstruction. 

The sloping lines in Fig. 1 could be drawn at any angle other 

6 



OBLIQUE PROJECTION 



than 30°. In Fig. 2, the same box is drawn with a 60° 

tion. It will be seen, in this latter case, that the inside 

of the box is also shown 

prominently. It is customary 

in the appUcation of this type 

of drawing, to use either 30°, 

45° or 60° for the slope, as these 

lines can be easily drawn with 

the standard triangles used in 

the drafting room. 



inclina- 
bottom 



202. Oblique projection of 
lines parallel to the plane of 
projection. In developing the 
theory, let XX and YY, Fig. 3, be 
two planes at right angles to each 
other. Also, let ABCD be a thin 
rectangular plate, the plane of Fig. 2. 

which is parallel to the plane XX. 

Suppose the eye is looking in the direction Aa, inclined* to the 
plane XX. Where this line of sight from the point A on the 





object appears to pierce or impinge on the piano XX, locate the 
point a. From the point B, assume that the eye is again directed 

* The ray must not be perpendicular, as this makes it an orthoKraphic 
projection. The ray cannot be parallel to the plane of projection, because 
it will never meet it, and, hence, cannot result in a projection. 



8 PARALLEL PROJECTING-LINE DRAWING 

toward the plane XX, in a line that is parallel to Aa; this 
second piercing point for the point B in space will appear 
at b. Similarly, from the points C and D on the object, the 
piercmg points 'on [the plane wdll be c and d, Cc and Dd being 
parallel to Aa. 

On the plane XX, join the points abed. To an observer, 
the figure abed will give the same mental impression as ^nll 
the object ABCD. In other words, abed is a drawing of the 
thin plate ABCD. ABCD is the objeet in space; abed is the 
corresponding oblique projeetion of ABCD. The plane XX is 
the plane of projeetion; Aa, Bb, Ce, and Dd, are the projeeting 
lines, making any angle with the plane of projection other than 
at right angles or parallel thereto. The plane YY serves the 
purpose of throwing the plane XX into stronger relief and has 
nothing to do with the projection. 

It will be observed that the figure whose corners are the 
points ABCDabed is an oblique rectangular prism, the opposite 
faces of which are parallel because the edges have been made 
parallel by construction. From the geometry, all parallel plane 
sections of the prism are equal, hence abed is equal to ABCD, 
because the plane of the object ABCD was originally assumed 
parallel to XX, and the projection abed lies in the plane XX. 
As a corollary, the distance of the object from the plane of pro- 
jection does not influence the size of the projection, so long as 
the plane of the object is continually parallel to the plane of 
projection. 

Indeed, any line, whether straight or curved, when parallel 
to the plane of projection has its projection equal to the line 
itself. This is so because the curved line may be considered 
as made up of an infinite number of very short straight lines. 

203. Oblique projection considered as a shadow. Another 
way of looking at the projection shown in Fig. 3 is to assume 
that light comes in parallel lines, obUque to the plane of pro- 
jection. If the object is interposed in these parallel rays, then 
abed is the shadow of ABCD in space, and thus presents an 
entirely different standpoint from which to consider the nature 
of a projection. Both give identical results, and the latter is 
here introduced merely to reenforce the imderstanding of the 
nature of the operation. 



OBLIQUE PROJECTION 9 

204. Oblique projection of lines perpendicular to the 
plane of projection. Let XX, Fig. 4, be a transparent plane 
surface, seen edgewise, and ab, an arrow perpendicular to XX, 
the end a of the arrow lying in the plane. Suppose the eye is 
located at r so that the ray of light rb makes an angle of 45° 
with the plane XX. If all rays of light from points on ab are 



\i^"-<5 



\ 



to ^P 



H^^ 



X 
Fig. 4. 




Fig. 5. 



parallel to rb, they will pierce the plane XX in a series of points, 
and ac, then, will become the projection of ab on the plane XX. 
To an observer standing in the proper position, looking along 
lines parallel to rb, ac will give the same mental impression as 
the actual arrow ab in space; and, therefore, ac is the projection 
of ab on the plane XX. The extremities of the line ab are hence 
projected as two distinct points a and c. Any intermediate 
point, as d, will be projected as e on the projection ac. 

From the geometry, it may be noticed that ac is equal to ab, 
because cab is a right angled triangle, and the angle acb equals 
the angle cba, due to the adoption of the 45° ray. Also, any 
limited portion of a line, perpendicular to the plane of projection, 
is projected as a line equal to it in leng-tli. The triangle cab may 
be rotated about ab as an axis, so that c describes a circle in 
the plane XX and thus ac will always remain equal to ab. This 
means that the rotation merely corresponds to a new position 
of the eye, the inclination of the ray always remaining 45° with 
the plane XX. 



10 



PARALLEL PROJECTING-LINE DRAWING 



The foregoing method of representing 45° rays is again shown 
as an oblique projection in Fig. 5. Two positions of the ray 
are indicated as rb and sb; the corresponding projections are 
ac and ad. Hence, in constructing obHque projections, the 
lines that are parallel to the plane of projection are drawn with 
their true relation to each other. The lines that are perpendicular 
to the plane of projection are drawn as an inclined line of a length 
equal to tke line itself and making any angle with the horizontal, 




Fig. 6. 



at pleasure. Here, again, the plane YY is added. The line of 
intersection of XX and YY is perpendicular to the plane of 
the paper, and is shown as a sloping line, because the two planes 
themselves are pictured in obhque projection.* 

205. Oblique projection of the combination of parallel 
and perpendicular lines to the plane of projection. Fig. 
6 shows a box and its projection, pictorially indicating all the 
mental steps required in the construction of an oblique pro- 
jection. The object (a box) A is shown as an oblique projection; 

* Compare this with Figs. 1 and 2. The front face of the box is shown 
as it actually appears, because it is parallel to the plane of projection (or 
paper). The length of the box is perpendicular to the plane of the paper 
and is projected as a sloping line. 



OBLIQUE PROJECTION 



11 



its projection B on the plane of projection appears very much 
distorted. This distortion of the projection is due to its being 
an oblique projection, initially, which is then again shown in 
oblique projection. From what precedes, the reader should find 
no difficulty in tracing out the construction. Attention may 
again be called to the fact that the extremities of the lines per- 
pendicular to the plane of projection are projected as two distinct 
points. 



206. Oblique projection of circles.* When the plane of a 
circle is parallel to the plane of projection, it is drawn with a 
compass in the ordinary way, because the projection is equal 
to the circle itself (202). When the plane of the circle is per- 
pendicular to the plane of 
projection, however, it is 
shown as an ellipse. Both 
cases will be illustrated by 
Fig. 7, which shows a cube 
in obHque projection. In 
the face abed, the circle is 
shown as such, because the 
plane of the circle is par- 
allel to the plane of pro- 
jection, which, in this case, 
is the plane of the paper. It 
will be noticed that the circle 
in abed is tangent at points 

midway between the extremities of the lines. If similar points 
of fcangency are laid off in the faces aefb and fbeg and a smooth 
curve be drawn through these points, the result will be an ellipse; 
this ellipse is, therefore, the obHque projection of a circle, whose 
plane is perpendicular to the plane of projection. The additional 
lines in Fig. 7 show how four additional points may be located 
on the required ellipse. 

It may be shown that a circle is pr()j(H't(Ml as an ellipse in all 
cases except when its plane is parallel to tlu^ plan(^ of projection, 
or, when its plane is chosen parallel to the ])roj(M*ting lines. In 
the latter case, it is a line of a length (^lual to the dianu^ter of the 

* When projecting circlos in ixu'iHMulicular planes, (lio 'M^ slopt* oHVrs 
nn advantage because the ellipse is easily approximatiHl. See Art. 405. 




Fig. 7. 



12 



PAEALLEL PKOJECTING-LINE DRAWING 



circle. The reason for this Tvdll become evident later in the 
subject. (It is of insufficient import at present to dwell on it 
at length.) 

207. Oblique projection of inclined lines and angles. 

At times, lines must be drawn that are neither parallel nor 
perpendicular to the plane of projection. A reference to Fig. 
8 will show how this is done. It is desired to locate a hole in 
a cube whose edge measures 12". The hole is to be placed in 
the side bfgc, 8" back from the point c and then 4" up to the 
point h. To bring this about, lay off ck = 8" and kh (vertically) 

= 4" and, then h is the 

T^Z J required point. Also, hkc 

is the oblique projection 
of a right angled tri- 
angle, whose plane is per- 
pendicular to the plane 
of projection. Suppose, 
further, it is desired to 
locate the point m on 
the face abfe, 7" to the 
right of the point a and 
d" back. The dimensions 
show how this is done. 
Again, man is the ob- 
lique projection of aright 
angled triangle, whose 
legs are 5" and 7". This method of lajdng off points is virtually 
a method of offsets.* The point m is offset a distance of 5" 
from ab; likewise h is offset 4" from eg. If it be required to lay 
off the diagonal of a cube, it is accomplished by making three 
offsets from a given point. For instance, consider the diagonal 
ce. If c is the starting point, draw eg perpendicular to thft 
plane (shown as an inclined line), then fg, vertically upward, 
and finally fe, horizontally to the left; therefore, ee is the diagonal 
of the cube, if eg = gf = fe.t 

* This method is of importance in that branch of mathematics known 
as Vector Analysis. Vectors are best drawn in space by means of oblique 
projection. 

t If two given lines are parallel in space, their obhque projections are 
parallel under any conditions. The projecting lines from the extremities of 




Fig. 8. 



OBLIQUE PROJECTION 



13 



A word may be said in reference to round holes appearing 
in the oblique faces of a cube. As has been shown, circles are 
here represented as ellipses (206), but if it were desired to cut 
an eUiptical hole at either h or m, then their projections would 
not give a clear idea of the fact. Such cases, when they occur, 
must be covered by a note to that effect; an arrow from the 
note pointing to the hole would then indicate, unmistakably, 
that the hole is to be drilled (for a round hole), otherwise its 
shape should be called for in any way that is definite. It seems, 
therefore, that obUque projection cannot fulfill the needs of 
commercial drawing in every respect; and, indeed, this is true. 
Other methods also have certain advantages and will be treated 
subsequently, t 

208. Representation of visible and invisible lines. While 
viewing an object, the observer finds that some lines on the 
object are visible. These fines are drawn in full on the projection. 
There are, however, other lines, invisible from the point of view 
chosen and these, when added, are shown dotted. Fig. 9 shows 
all the visible and invisible 
lines on a hollow circular 
cylinder. Dimensions are 
appended and the cylinder 
shaded so that no question 
should arise as to its identity. 
It can be observed that the 
drawing is clear in so far as 
it shows that the hole goes 
entirely through the cylin- 
der. Were the dotted lines 
omitted, one could not tell 

whether the hole went entirely through, or only part way 
through. Hence, dotted lines ma> add to the clearness of a 
drawing; in such cases they should be added. At times, how- 
ever, their addition may lead to confusion; and, then, only the 

tho p;ivon lines determine planes that cut the plane of projection in lines 
which are the projections of the given lines. The projecting planes from 
the given lines are parallel, and, hence, their projections are j)arallel, since 
it is the cas(i of two paralh^l planers (uit by a third plane. 

t The student will obtain many suggestions by copying such siniiilc 
illustrations as Figs. 1, 2, 7, 8 and 9. 




Fig. 9. 



14 PARALLEL PROJECTING-LINE DRAWING 

more important dotted lines added, and such others, considered 
unnecessary, should be omitted. Practice varies in this latter 
respect and the judgment of the draftsman comes into play 
at this point; ability to interpret the dra^ving rapidly and 
accurately is the point at issue. 

209. Drawings to scale. Objects of considerable size cannot 
be conveniently represented in their full size. The shape is 
maintained, however, by reducing the length of each a definite 
proportion of its original length, or, in other words, b}^ drawing 
to scale. Thus, if the drawing is one-half the size of the object, 
the scale is Q" = 1 ft. and is so indicated on the dra^vdng by a note 
to that effect. The scales in common use are 12" = 1 ft. or full 
size; 6'' = 1 ft. or half-size; 3" = 1 ft. or quarter size; W; I" 

r; ¥'; I"; i"; A"; ¥'; A''; A"; etc. = i ft. The smaller 

sizes are used for ver}^ large work and \4ce versa. In railway 
work, scales like 100' = 1 inch, or 10000' = ! inch are common. 
In watch mechanism, scales like 48" = 1 ft. or " four times actual 
size " or even larger are used, since, other^^se, the drawings 
would be too small for the efficient use of the workman. Irre- 
spective of the scale used, the actual dimensions are put on the 
drawing and the scale is indicated on the draTvdng by a note 
to that effect. 

If some dimensions are not laid out to the scale adopted, 
the drawing may create a ^Tong impression on the reader and 
this should be avoided if possible. When changes in dimen- 
sions occur after the completion of a drawing and it is im- 
practicable to make the change, the dimension may be 
underlined and marked conveniently near it NTS, meaning ^'not 
to scale." 

210. Examples of oblique projection. Fig. 10 shows a 
square block "v\dth a hole in its centre. The dimension lines indicate 
the size and the method of making the drawing when the planes 
of the circles are chosen parallel to the plane of the paper (plane 
of projection). The circle in the visible face is drawn with a 
compass to the desired scale. The circle in the invisible face 
(invisible from the point of view chosen) is di'awn to the same 
radius, but, its centre is laid off on an inclined line, a distance 
back of the visible circle, equal to the thickness of the block. 
The circle in the front face is evidently not in the same plane 



OBLIQUE PROJECTION 



15 



as that in the distant face. The Hne joining their centres is 
thus perpendicular to the plane of projection, and is, hence, 
laid off as an inclined line. 







^ \ 








y>^ 


A 


1 






M 




1 

1 


^ 




I 


^ 


, 


I 




\,/A 


/ 


» 






V 




V- 


v>*'' 




csi 




! N 

1 

1 


^ 


-^ 








^^'' 






^ 








■p^ 




j>A 




, 2" 




< 2" 








' 









Fig. 10. 




Fig. 11. 



When the plane of the circles is made perpendicular to the 
plane of projection, the circles are projected as ellipses. Fig. 
11 shows how the block of Fig. 10 is drawn when such is the 



16 



PARALLEL PROJECTING-LINE DRAWING 



case. It is to be observed, that the bounding square is to be 
drawn first and then the elHpse (projection of the circle) is inscribed. 
When the circles in both faces are to be showTi, the bounding 




Fig. 13. 



square must be replaced by a bounding rectangular prism. This 
rectangular prism is easily laid out and the ellipses are inserted 
in the proper faces. The method of using bounding figures of 



OBLIQUE PROJECTIOX 



17 



simple shape is of considerable importance when applying the 
foregoing principles to obhque projection. 




Fig. 14. 




Fig. 15. 



Fig. 12 is another illustration of an object, differing from 
Figs. 10 and 11 in so far as the hole does not go entirely through 



18 



PARALLEL PROJECTING-LINE DRAWING 



from face to face. The centres for the different circles are found 
on the axis. The distances between the centres, measured on 
the inchned Hne, is equal to the distances between the planes 
of the corresponding circles. Since the circles are drawn as such, 
the planes must, therefore, be parallel to the plane of projection. 
The axis of the hole is perpendicular to the plane of projection, 
and, hence, is projected as an inclined line. 

The object in Fig. 12 is also sho^ni with the planes of the 
circles perpendicular to the plane of projection in Fig. 13. Every 
step of the construction is indicated in the figure and the series 
of bounding prisms about the cylinders is also shown. 




Fig. 16. 



A somewhat different example, showing the necessity of 
bounding figures, is given in Fig. 14. The bounding figure, 
shown in Fig. 15, is laid out as given and it becomes a simple 
matter to insert the object subsequently. It should be noted 
how the rectangular projection becomes tangent to the cjdinder 
and that the only way to be certain of the accuracy of the 
draT\dng, is to use these bounding figures and make mental 
record of the relative location of the lines that make up the 
drawing. 

If objects are to be dra^vTi whose lines are inchned to each 
other, the principles so far developed offer simple methods for 
their presentation. Fig. 16 shows a tetrahedron with a bounding 



OBLIQUE PROJECTION 



19 



rectangular prism. The apex is located on the top face and 
its position is determined from the geometric principles imposed. 
Solids, as represented in the text-books on geometry, are drawn 
in this way. Some confusion may be avoided by observing 
that angles are only preserved in their true relation in the planes 
parallel to the plane of projection (207). 

The concluding example of this series is given in Fig. 17. 
It is known as a bell-crank and has circles shown in two planes 
at right angles to each other, The example furnishes the clue 




Fig. 17. 



to constructing any object, however complicated it may be. 
Base lines ab and be are first laid out to the required dimensions 
and to the desired scale. In this example, the base lines are 
chosen parallel to the plane of projection, and hence are projected 
as a right angle, true to dimensions. The thickness of the two 
lower cylinders is laid off as an inclined line from each side of the 
base line and the circles are then drawn. The upper circles 
(shown as ellipses) may need some mention. A bounding rec- 
tangular prism is first drawn, half of which is laid off on each 
side of the base line (true in this case but it may vary in others). 



20 



PARALLEL PROJECTIXG-LIXE DRAWING 



The circles and inclined lines are filled in after the guiding details 
are correctly located. This drawing may present some difficulty 
at first, but a trial at its reproduction vrill reveal no new prin- 
ciples, only an extreme apphcation. 

On completion of the drawing, the bomiding figm-e may be 
removed if its usefulness is at an end. When inclined lines 
appear frequently on the dra^dng, the bounding figures can be 
made to serve as dimension lines, and so help in the interpre- 
tation. The draftsman must determine what is best in each 
case, remembering, always, that the drawing must be clear 
not only to himself, but to- others who may have occasion to 
read it. 

211. Distortion of oblique projection. A view of a com- 
pleted machine suffers considerable distortion when drawn in 



M.. 



X 

Fig. 18. 



*-i 




Fig. 19. 



oblique projection because the ej^e cannot be placed in any one posi- 
tion, whereb}^ it can view the drawing in the manner the projec- 
tion was made.* To overcome this difficulty to some extent and 
to avoid bringing the distortion forcibh^ to the attention of the 
observer, the projecting lines can be so chosen that the per- 
pendicular to the plane of projection is projected as a shorter 
line than the perpendicular itself. Fig. 18 shows this in con- 
struction. XX is a vertical transparent plate, similar to that 
shown in Fig. 4. The ra}^ rb makes an angle vAih. XX greater 
than 45°, and, by inspection, it is seen that the projection of 
ab on XX is ac, which is shorter than ab, the perpendicular. 

The application of the foregoing reduces simply to this: All 
lines and curves parallel to the plane of projection are shoTvn 
* This condition is satisfied in Perspective Projections. 



OBLIQUE PROJECTION 21 

exactly the same as in oblique projection with 45° ray incUnation. 
The lines that are perpendicular to the plane of projection are 
reduced to i, i, i, etc. of their original length and reduced or 
increased to the scale adopted in making the dra^ving. This 
mode of representation* is suitable for making catalogue cuts 
and the like. It gives a sense of depth without very noticeable 
distortion, due to two causes: the impossible location of the 
eye while viewing the drawing, and, the knowledge of the apparent 
decrease in size of objects as they recede from the eye. A single 
illustration is shown in Fig. 19. 

212. Commercial application of oblique projection. 

Oblique projection is useful in so far as it presents the three 
dimensions in a single view. When curves are a part of the 
outline of the object, it is desirable to make the plane of the 
curve parallel to the plane of projection, thereby making the 
]:)rojection equal to the actual curve and also economizing time 
in making the drawing. Sometimes it is not possible to carry 
this out completely. Fig. 17, already quoted, shows an example 
of this kind. It is quite natural to make the drawing as shown, 
because the planes of most of the circles are parallel to the plane 
of projection, leaving, thereby, only one end of the bell-crank 
to be projected with ellipses. 

Oblique projections, in general, are perhaps the simplest 
types of drawings that can be made, if the objects are of com- 
paratively simple shape. They carry with them the further 
advantage that even the uninitiated are able to read them, when 
the objects are not unusually intricate. The making of oblique 
projections is simple, but, at the same time, they call on the 
imagination to some extent for their interpretation. This is 
largely due to the fact that the eye changes its position for each 
point projected, and that no one position of the eye will properly 
place the observer with respect to the object. 

The application of oblique projection to the makhig of 
drawings for solid geometry is already known to the student 
and the resulting clarity has been noticed. Otlier types of i^ro- 
jections have certain advantages which will be considered in (hie 
order. 

* This type of projection has been called Pseudo PersptM'livc by Dr. 
MucCord in his Descriptive (Jeonietry. 



22 PARALLEL PROJECTIXG-LIXE DRAWING 

The convenience of oblique projection to the lajdng out of 
piping diagrams is worthy of mention. Steam and water pipes, 
plumbing, etc., when laid out this way, result in an exceedingly 
readable dra\\dng. 

QUESTIONS OX CHAPTER II 

1. What is an obhque projection? 

2. What is a plane of projection? 

3. AVhat is a projecting line? 

4. Prove that when a rectangle is parallel to the plane of projection the 

projection of the rectangle is equal to the rectangle itself. 

5. Does th£ distance of the object from the plane have any influence on 

the size of the projection? Why? 

6. Prove that any hne, whether straight or curved, is projected in its 

true form when it is parallel to the plane of projection. 

7. Show under what conditions a projection may be considered as a 

shadow. 

8. Prove that when a hne is perpendicular to the plane of projection, 

it is projected as a line of equal length, when the projecting rays 
make an angle of 45° ^'ith the plane of projection. L^se a diagram. 
9; Prove that any hmited portion of a hne is projected as a hne of 
equal length, when the hne is perpendicular to the plane of projec- 
tion and the projecting hnes make an angle of 45° \^ith the plane 
of projection. 

10. Show how a perpendicular may be projected as a longer or a shorter 

hne, if the angle of the projecting hnes chffers from 45°. 

11. Why can not the projecting lines be selected parallel to the plane of 

projection? 

12. Show that when a hne is parallel to the plane of projection, it is 

projected as a. hne of equal length, irrespective of the angle of the 
projecting lines, pro^^ded the projecting hnes are inchned to the 
plane of projection. 

13. Wh}^ may the slope of the projection of a hne perpendicular to the 

plane of projection be dra^n at any angle? 

14. Prove that when two hnes are parallel to each other and also to the 

plane of projection, their projections are parallel. 

15. Prove that when two hnes are perpendicular to the plane of pro- 

jection, their projections are parallel to each other. 

16. Draw two rectangular planes at right angles to each other so that the 

edges of the planes are parallel or perpendicular to the plane o,^ 
the paper (or projection). 

17. Draw a cube in obhque projection and show wliich hnes are assumed 

parallel to the plane of projection and wliich hnes are perpendicular 
to the plane of projection. 

18. Draw a cube in oblique projection and show how the circles are 

inserted in each of the ^-isible faces. 



OBLIQUE PROJECTION 



23 



19. Show how angles are laid off on the face of a cube in obUque pro- 

jection. 

20. Under what conditions is the angular relation between hnes pre- 

served? 

21. Draw a hne that is neither parallel nor perpendicular to the plane 

of projection. (Use the cube, in projection, as a bounding 
figure.) 

22. Prove that any two hnes in space are projected as parallels when 

they themselves are paraUel. 

23. Under what conditions will the obhque projection of a hne be a 

point? 

24. How are visible and invisible hnes represented on a drawing? 

25. What is meant by drawing to scale? 

26. Why is it desirable to have all parts of the same object drawn to 

true scale? 











^^ 






X 


X 




X 


1" 




< 2". 








• 6"^ 








" 













Fig. 2A. 



Fig. 2B. 



28 



29 



30 



27. What considerations govern the choice of the scale to be used on a 

drawing? 
Show how the distortion of an oblique projection may be reduced by 

changing the angle of the projecting lines. 
Why is it impossible to locate the eye in one position and view the 

projection in the manner in which it was made? 
Draw a rectangular box with a hinged cover, in obhque projection, 

and show the cover partly raised. 

31. Draw a lever, having a round hole on one end so as to fit over a 

shaft. Have the plane of the circles parallel to the plane of 
projection. 

32. Draw an oblong block, 2"x3"xr)" long, in oblique projection, 

having a 1" hole in its centre, 4" deep. 

33. Draw a cylindrical shaft, 6" in diameter and 18" long, in oblique 

projection, having a rectangular hole, 2"X3"x5" deoj), from 
each end. Lay out to a scale of 3" = 1 ft. and aiiix all dimensions. 



24 PARALLEL PROJECTING-LINE DRAWING 

<i4. Draw a triangular prism, in oblique projection, showing how the 
bounding figure is used. 

35. Draw a hexagonal prism, in obhque projection, and show all the 

invisible (dotted) hues on it. 

36. Draw a hexagonal pyramid in obhque projection. 

37. Make a material hst for the box shown in Fig. 2A. 

38. Draw the circular cyhnder, 4" diameter and 6" long, shown in 

Fig. 2B. On the surface of this cyhnder, two semi-circular grooves 
are cut, as shown by the dimensions. Make two drawings in 
obhque projection, one sho\\dng the plane of the circles parallel 
to the plane of projection and the other, \vith the plane of the 
circles perpendicular to the plane of projection. 

Note. — For additional drawing exercises see examples in Chapters 
III and IV. 






CHAPTER III 



ORTHOGRAPHIC PROJECTION 



301. Nature of orthographic projection. Take, for 



example, a box 6''X12''X4" high, made of wood. 



thick. 



This box is shown orthographically in Fig. 20, and requires two 
distinct views to illustrate it properly. The upper view, or 
elevation, shows the side of the box whose outside dimensions 
are 4'' X 12", while the thickness of the wood is indicated by the 




-^# 



TT 



Sl_ 



o 



o 



n 



o 
o 



! I 



Fig. 20. 



Fig. 21. 



dotted lines. The lower view is called the plan, and is obtained 
by looking down into the inside of the box. It is thus to 
be remembered that the two views are due to two distinct 
directions of vision on the part of the observer. 

As another example of this mode of representation, consider 
the object shown in Fig. 21. It is here a rectangular plate, 
5"XS" and 1" thick, with a square hole in its centre. The 
metal around the square hole projects Y' above the surface of 
the plate. In addition, there are four bolt holes which ena])le 
the part to be secured to a machine with bolts. As befort^ 
two views are shown with the necessary dimensions for con- 
struction. 

25 



26 



PARALLEL PROJECTING-LINE DRAWING 



302. Theory of orthographic projection. Let Fig. 22 
represent an oblique projection of two plane surfaces HH and 
W, at right angles to each other. For mechanical operations 
to be performed later, it is assumed that they are hinged at 
their intersection so that both planes may be made to lie as one 
flat surface, instead of two separate surfaces at right angles 
to each other. The plane HH, shown horizontally, is the hori- 
zontal plane of projection; that shown vertically, is the vertical 
plane of projection; their intersection is called the ground line. 
The two planes, taken together, are known as the principal planes 




Fig. 22. 



of projection. The object is the 4"X6"X12'' box chosen as 
an illustration in Fig. 20. The drawing on the horizontal plane 
is the horizontal projection; while that on the vertical plane is 
the vertical projection. 

The method of constructing the projection consists of dropping 
perpendiculars from the object upon the planes of projection. 
Thus, in other words, the projecting lines are perpendicular to 
the plane of projection. To illustrate: The box is so located 
in space that the bottom of it is parallel to the horizontal plane 
(Fig. 22) and the 4"X12" side is parallel to the vertical plane. 
From the points A, B, C, and D, perpendiculars are drawn to 
the vertical plane and the points, where these perpendiculars 
pierce or impinge on the plane, are marked a', b', c', and d', to 



ORIHOGRAPHIC PROJECTION 27 

correspond with the similarly lettered points on the object. 
By joining these points with straight lines, to correspond with 
the lines on the object, the vertical projection is completed, 
when the dotted lines showing the inside of the box are added. 

Turning to the projection on the horizontal plane, it is seen 
that A, B, E, and F are the corners of the box in space, and 
that perpendiculars from these points to the horizontal plane 
determine a, b, e, and f as the horizontal projection. It is 
assumed that the observer is looking down on the horizontal 
plane and therefore sees the inside of the box; these hues are 
hence shown in full, although the projecting perpendiculars are 
omitted so as to avoid too many lines in the construction. 

303. Revolution of the horizontal plane. It is manifestly 
impracticable to carry two planes at right angles to each other, 
each containing one projection of an object. A more con- 
venient way is to represent both projections on a single plane 
surface, so that such drawings can be represented on a flat sheet 
of paper. The evident expedient, in this case, is to revolve 
the horizontal plane about the ground line as an axis, until it 
coincides with the plane of the vertical plane. The conventional 
direction of rotation is shown by the arrow in Fig. 22, and to 
accomplish this coincidence, a 90° revolution is required. In 
passing, it may be well to note that it makes no difference whether 
the horizontal plane is revolved as suggested, or whether the 
vertical plane is revolved in the opposite direction into coin- 
cidence with the horizontal plane. Both accomplish the same 
purpose, and hence either method will answer the requirements. 

304. Position of the eye. The perpendicular projecting lines 
drawn to the planes of projection correspond ^vith a line of sight 
that coincides with these perpendiculars. Each point found on 
the projection, corresponds to a new position of the eye. All 
projecting lines to one plane are then evidentl}- parallel Ix^'ause 
they are all perpendicular to the same plane. As two projec- 
tions are required, two general directions of vision are necessary. 
That for the horizontal projection requires the eye aboA'e that 
plane, continually directed perpendicularly against it; thus the 
eye is continually shifting in position, although the direction of 
vision is fixed. Also, the vertical projection requir(\s that tlu^ 
eye be directed perpendicularly against it, but in this case, tiio 



28 



PARALLEL PROJECTIXG-LIXE DRAWING 



line of sight is perpendicular to that required for the horizontal 
projection. 

305. Relation of size of object to size of projection. 

The object is projected on the planes by lines perpendicular 
to it. If the plane of the object is parallel to the plane of pro- 
jection, then the projection is equal to the object in magnitude. 
This is true because the projecting lines form a right prism and 




'I y/ I 




Fig. 23. 



all the parallel plane sections are the same (compare with 202). 
Fig. 23 gives the construction of the projection in Fig. 21. 
ABCDa'b'c'd' is such a right prism because the plane of the object 
is parallel to the plane of projection and tlie projecting Unes 
are perpendicular to the plane of projection. 

306. Location of object with respect to the planes of 
projection. For purposes of draT\ing. the location of the object 
to the planes of projection is absolutely immaterial. In fact, 



ORTHOGRAPHIC PROJECTION 29 

the draftsman intuitively makes the projections and puts corre- 
sponding projections as close as is necessary to economize room 
on the sheet. 

307. Location of projections with respect to each other. 

In Figs. 20 and 21, the vertical projection is placed directly 
above the horizontal projection. Reference to Fig. 23 will show 
why such is the case. When the horizontal plane is revolved 
into coincidence with the vertical plane, the point d will describe 
the arc of a circle dd" * which is a quadrant; d" is the ultimate 
position of the point d after revolution, and must be on a line 
d'd'' which is perpendicular to the original position of the hori- 
zontal plane. So, too, every point of the horizontal projection 
is located directly under the corresponding point in the vertical 
projection, and the scheme for finding its position is identical 
to that for finding d at d". 

308. Dimensions on a projection. When the principal planes 
of the object are turned so that they are parallel to the planes of 
projection, then the edges will, in the main, be perpendicular 
to the planes of projection. In Fig. 23, DF is one edge of the 
object and it is perpendicular to the vertical plane of projection 
W. The projection of this line is d', because the projecting 
perpendicular from any point on DF will coincide with DF itself. 
The result of this is that the thickness of the object is not shown 
when the length and breadth are shown, or, in other words, only 
two of the three principal dimensions are shown in a single view. 
Thus, another view is required to show the thickness. If DC 
be considered a length, and DF a thickness, the horizontal pro- 
jection shows both as do and df. The vertical projection does 
not show the thickness DF as it is perpendicular to the vertical 
plane of projection. Hence, in reading orthographic projections, 
both views must be interpreted simultaneously, as each shows 
but two of the three principal dimensions and only one of i\w 
three is common to both projections. 

309. Comparison between oblique and orthographic 
projections. It is of interest here to show wherein xhv ortlio- 
graphic projection differs from the oblique. When Xhv phuw 

* d' is read d primo; d" is read d second; d'" is read d third; ami so 
on. 



30 



PARALLEL PROJECTIXG-LIXE DRAWING 



of the object is parallel to the plane of projection, the projection 
on that plane is equal to the object, whether it is projected 
orthographically or obliciuely. When a line is perpendicular to 
the plane of projection its extremities have two distinct pro- 
jections in oblique projection, but only one in orthographic pro- 
jection. This latter statement means simply that if the projecting 
lines instead of being oblique to the plane of projection, gradually 
assume the perpendicular position, the two projections of the 
extremities of any line approach each other until they coincide 
when the projecting lines are perpendicular. Therefore, in 
orthographic projection, the third dimension vanishes and a 
new vievr must be made in addition to the other, in order to 
represent a solid. 

310. Orthographic projection considered a shadow. The 

horizontal and the vertical projections may be considered as 
shadows on their respective planes. The source of light must 
be such that the rays emanate in paraUel lines, and are directed 
perpendicularly to the planes of projection. Evidently, the 
two views are due to two chstinct positions of the source of light, 
one whose rays are perpendicular to the horizontal plane while 
casting the horizontal shadow, and the other, whose rays are 
perpendicular to the vertical plane while casting the vertical 
shadow. 



311. Profile plane. Let A. 

projection and B. the vertical 





— 


J B 





in Fig. 24, be the horizontal 
projection of an object. The 
two views are identical, and to 
one unfamiliar with the object, 
they are indefinite, as it is im- 
possible to tell whether they are 
projections of a cylinder or of a 
prism. By the addition of either 
\4ew C or D, it is at once appar- 
ent that the object in question 
is a circular cylinder, a hole 
running part way through it 
and "^ith one end square. 
Fig. 25 shows how this profile is made. As customary, the 
horizontal and vertical planes are present and the projection 
on these planes should now require no further mention. A 



±- 



Fig. 24. 



ORTHOGRAPHIC PROJECTION 



31 



profile plane (or end plane as it may be called) is shown on the 
far side of the object and is a plane that is perpendicular to 
both the horizontal and vertical planes (like the two adjacent 




Fig. 25. 

walls and the floor of a room meeting in one corner). A series 
of perpendiculars is dropped from the object upon this profile 






B 


- 


"■ 










Fig. 26. 

plane, as shown by the dotted lines, and thus the side view is 
determined. 



312. Location of profiles. If the profile view is to show 
the object as seen from the left side, it is put on the left side of 
the drawing, and vice versa. Fig. 24 shows two profile views 
located in accordance with this direction. Either views B and C 



32 



PARALLEL PROJECTIXG-LINE DRAWING 



or B and D completely represent the object. In this case, 
although this is not always so, the horizontal projection is not 
essential. 

Fig. 26 gives still another illustration of an object that is 
not as sjTnmetrical as that immediately preceding. The illus- 
tration is chosen to show exactly how the profile planes are 
revolved into the vertical plane, if the vertical plane be assumed 
as the plane of the paper. A is the horizontal and B the vertical 
projection of the object. C and D are two profiles, drawn against 
the vertical projection, whereas E is a profile drawn against 
the horizontal projection. Fig. 27 shows a plan \dew of the 
vertical and two profile planes. In reading this drawing, the 



V 



V 
R 



^ 



Fig. 2', 



horizontal plane is the plane of the paper, while the vertical plane 
is seen on edge and is shown as W, as are also the left and right 
profile planes indicated respectively as LL and RR. 

In making the projection on the horizontal plane, the object 
is above the plane and the projecting perpendiculars are dropped 
from points on the object to the horizontal plane, which in this 
case is the plane of the paper. The construction of the vertical 
projection (that on W) is indicated by the arrow A. The 
arrangement here shoT\Ti corresponds to the \dews A and B in 
Fig. 26. 

When making the profile projections, the planes are assumed 
as transparent, and are located between the object and the 
observer. As the observer traces the outline on these profile 
planes, point by point, each ray being perpendicular to the plane, 
the resultant picture so drawn becomes the required projection. 
If, then, the planes LL and RR be revolved in the direction of the 



ORTHOGRAPHIC PROJECTION 



33 



arrows until they coincide with the vertical plane, and then the 
vertical plane be further revolved into the plane of the paper, 
the final result will be that of Fig. 26 with view E omitted. 

View E is a profile drawn against the horizontal projection 
and is shown on the deft because it is the projection on the profile 
plane LL. It has been revolved into the horizontal plane, by 
revolving the profile plane so that the upper part of the plane 
moves toward the object into coincidence with the horizontal 
plane. 

Fig. 26 has more views than are necessary to illustrate the 
object completely. In practice, all would not be drawn, their 
presence here is necessary only to show the method. 

313. Section plane. The addition of dotted lines to the 
drawing of complicated objects is unsatisfactory at times on 
account of the resultant confusion of lines. This difficulty can 
be overcome by cutting the object by planes, known as section 
planes. The solid material when 
so exposed is sectioned or cross- 
hatched by drawing a series 
of equidistant lines over the 
exposed area. A convenient 
mnemonic in this connection is 
to assume that the cut is made 
by a saw and that the resultant 
tooth marks represent the sec- 
tion lines. Fig. 28 shows what 

is known as a stuffing box on a steam engine. This is a special 
case where but one projection is shown in section and one profile. 
The left-hand view might have been shown as an outside view, 
but the interior lines would then have been shown dotted. As 
it is, the object is cut by the plane ab and this half portion is 
shown to the left, sectioned of course, because the cut is not 
actual. 

Another example is seen in Fig. 29, where a fly-wheel is 
represented in much the same way as in the illustration in Fig. 
28. It differs somewhat from that immediately preceding in 
so far as the two views do not have the theoretical relation. 
Were the wheel actually cut by the plane ab then the arms 
(spokes) shown in the profile would have to be sectioned. As 




Fig. 28. 



34 



PARALLEL PROJECTING-LINE DRAWING 



shown, however, the arms appear in full as though the section 
plane passed through the wheel a short distance ahead of the 
spokes. The convention is introduced for a double purpose: 
In the first place it avoids peculiar projections as that for the 
plane cd for instance, where the spokes would be foreshortened 
because they incline to the plane of projection. In the second 
place, the sectioning of the spokes is the conventional method 
of showing a band wheel,* that is, a wheel with a solid web, 
or, in other words, without spokes. Hence, it appears that 
although it may not seem like a rational method of drawing, 
still the attending advantages are such as make it a general 
custom. The mechanics who use the drawings understand this, 
and therefore it becomes common practice. 




Fig. 29. 



Many more examples could be added, but they would be 
too complicated to be of illustrative value. It may be said, that, 
in some cases, six or more sections may be made to illustrate 
the object completely. They are located anywhere on the draw- 
ing and properly indicated, similar to cd in Fig. 29. It may also be 
mentioned, that in cases like that of the fly-wheel, the shaft 
is not cut by the section plane but is shown in full as it appears 
in Fig. 29. 

314. Supplementary plane. Fig. 30 shows a Y fitting used 
in pipe work for conveying steam, water, etc., and consists of 
a hollow cylindrical shell terminating in two flanges, one at 

* Fig. 43 is an example of a band wheel. 



ORTHOGRAPHIC PROJECTION 



35 



each end. From this shell there emerges another shell (in this 

case, smaller in diameter), also terminating in a flange. A is the 

Y fitting proper; B is the end view of one flange, showing the 

bolt holes for fastening to a 

mating flange on the next 

piece of pipe, not shoT\Ti. The 

view B shows only the one 

flange that is represented by 

a circle, because the profile 

plane is chosen so as to be 

parallel to that flange. If the 

flange at C be projected on 

this same profile plane, it 

would appear as an ellipse, 

and, as such, could not be drawn with the same facility as a 

circle. HerC; then, is an opportunity to locate another plane, 




Fig. 30. 




Fig. 31. 



called a supplementary plane, paralk^l to the flange at C. The 
projection of C on this supplenuMitary i)Iane will be a circU^ 
and is therefore readily drawn witli a compass. 



36 



PARALLEL PROJECTIXG-LIXE DRAWING 



Only one-half of the actual circle may be sho\sTi if desired 
so as to save time, space, or both time and space. The bolt 
holes are shown in the supplementary view at C just the same 
as in \'iew B. Either plane may be considered supplementary 
to the other: on neither plane is the entire projection made, 
because the object is of so simple a character. To avoid the 
possibility of any error arising, the supplementary projection 
is, if it is at all possible, located near the part to be illustrated. 
If, for any reason, this ^-iew cannot be so located, a note inchcating 
the proper position of the ^-iew is added to the drawing. 

315. Angles of projection. Up to the present point, no 
attention has been devoted to the angles in which the pro- 




o 








O ! O 




1st Angle 2nd Angle 3rd Angle 
1-iG. 32. 



4th Angle 



jections were made. As will soon appear, the examples so far 
chosen were all in the first angle. Fig. 31 shows two planes 
HH and W, intersecting at right angles to each other. The 
planes form fotir dihedral angles, nmnbered consecutively in 
a counter-clockwise manner as indicated. The same object is 
shown in all four angles, as are also the projections on the 
planes of projection, thus making four distinct projections. 

316. Location of obsener in constructing projections. 

The eye is always located above the horizontal plane in making 
any horizontal projection. That is. for objects in the first and 
second angles, the object is between the plane and the observer; 
for objects in the third and fourth angles the plane is between 



ORTHOGRAPHIC PROJECTION 37 

the object and the observer. While constructing the vertical 
projections the eye is always located in front of the vertical plane. 
That is, for objects in the first or fourth angles, the object 
is between the plane and the observer; but in the second and 
third angles, the plane is between the object and the observer. 
This latter means simply that the observer stands to the right of 
this vertical plane W and views it so that the line of sight is 
always perpendicular to the plane of projection. 

317. Application of angles of projection to drawing. 

If the horizontal plane be revolved about the ground line XY 
as indicated by the arrows, until it coincides with the vertical 
plane of projection, it will be seen that that portion of the hori- 
zontal plane in front of the vertical plane will fall below the 
ground line, whereas that portion of the horizontal plane behind 
the vertical plane will rise above the ground line. 

Supposing that in each position of the object in all four 
angles, the projections were made by dropping the customary 
perpendiculars to the plane of projection, and in addition, that 
the revolution of the horizontal plane is accomplished, then 
the resultant state of affairs will be as shown in Fig. 32. For 
purposes of illustration, the ground line XY is drawn although 
it is never used in the actual drawing of objects.* Also, the 
object has been purposely so located with respect to the planes 
that the second and fourth angle projections overlap. Mani- 
festly the second and fourth angles cannot be used in drawing 
if we wish to be technically correct. It may be possible so to 
locate the object in the second and fourth angles, by simply 
changing the distance from one plane or the other, that the 
two projections do not conflict, but a little study will show that 
the case falls either under first or third angle projection, depending 
upon whether the vertical projection is above the horizontal 
projection, or, below it. 

It will be seen that in the first angle of projection, the plan 
is below and the elevation is above; whereas in the third angle 
of projection the condition is reversed, that is, the plan is above 
and the elevation is below. Strictly speaking, the profile, section 
and supplementary planes, have nothing to do with the angle 

* When linos, points, and phin(\s an^ to be represented orlhofiraphically, 
the Rround line becomes a necessary adjunct. 



38 



PARALLEL PROJECTING-LINE DRAWING 



of projection, but it is quite possible to take a single projection 
with its profile, and locate it so that it corresponds to a third 
angle projection. Thus, there appears a certain looseness in 
the application of these principles. In general, the third angle 
of projection is used more than any other * as the larger number 




e- 




1st Angle 



3rd Angle 



Fig. 33. 



of mechanics are familiar with the reading of drawings in this 
angle. Fig. 33 shows the same object in the first and third 
angles of .projection. A profile, or end view, is also attached 
to each, thus making a complete, though simple illustration. 

318. Commercial application of orthographic projection. 

Orthographic projection is by far the most important method 
of making drawings for engineering purposes. Other types of 
drawing have certain advantages, but. in general, they are limited 
to showing simple objects, made up principally of straight lines. 

Some experience is required in reading orthographic projections 
because two or more views must be interpreted simultaneously. 
This experience is readily acquired by practice, both in making 
drawings and in reading the drawings of others. 
> To compensate for the more difficult interpretation of this 
type of drawing, there are inherent advantages, which permit 
the representation of any object, if it has some well defined 
shape. By the aid of sections, profiles, and supplementary 
planes, any side of a regular body can be illustrated at will, 
and further than this, the curves are shown with such peculiarity 
as characterizes them. Bodies, such as a lump of coal, or a spade 

* The third angle of projection should be used in preference to the first 
because the profile, section, and supplementary planes conform to third 
angle projection. 



ORTHOGRAPHIC PROJECTION 



39 



full of earth, are considered shapeless, and are never used in 
engineering construction. Even these can be represented ortho- 
graphically, however, although it is quite difficult to draw on 
the imagination in such cases. 

QUESTIONS ON CHAPTER III 

1. What are the principal planes of projection? Name them. 

2. What is the ground line? 

3. What angles do the orthographic projecting Hnes make with the 

plane of projection? 




\ 




- 


- 


- 1 

-1 








— 




1- 
/ 


— - 


- 


- 


-h 






/ 




- 





Fig. 3^. 



4. What is the horizontal projection? 

5. What is the vertical projection? 

6. Why is the horizontal plane revolved 90' 

jections? 



after making the pro- 




FiG. 3B. 



7. Could the vertical plane be revolved instead of the horizontal plane? 

8. Make a sketch of the planes of projection and show by arrow how 

the revolution of the planes is accomplished. 

9. How is the eye of the observer directed in making an orthographic 

projection? 



40 



PARALLEL PROJECTING-LINE DRAWING 



10. What is the general angular relation between the projecting lines 

to the horizontal plane and the vertical plane of projection? 

11. Wh}^ is the projection of the same size as the object? 

12. Why is the ground hne omitted in making orthographic projections? 

13. Wh}^ are corresponding projections located directly over each other? 

Show by diagram. 

14. Why does an orthographic projection show only two of the three 

principal dimensions of the object? 




15. Why must the two views of an orthographic projection be interpreted 

simultaneously? 

16. Compare the direction of the projecting hues of obUque projection 

with those of orthographic projection. 

17. Show how the source of Hght must be located in order that the 

shadow should correspond to a true projection. 




Fig. 3D. 

18. To cast the horizontal and vertical shadows, is it necessary to have 

two distinct positions for the source of hght? 

19. What is a profile plane? 

20. How is the profile plane located with respect to the principal planes 

of projection? 

21. How is the profile plane revolved into the plane of the drawing? 

Show by diagram. 



ORTHOGRAPHIC PROJECTION 



41 



22. How are the profiles located ^dth respect to the main projection? 

Show by some simple sketch. 

23. What is a section plane? 

24. When is a section plane desirable? 

25. How is the section constructed? 




Fig. 3^. 



26. How is the section located with respect to the main projection? 

27. Show a simple case where only one projection and one section com- 

pletely determine an object. 




Fig. SF. 



28. What is a supplementary plane? 

29. When is it desirable to use a supplementary plane? 

30. How is the supplementary projection located witli respect to the 

main projection? 



42 



PARALLEL PROJECTING-LINE DRAWING 



3L Show why the profile plane is a special case of the supplementary 

plane. 
32. Make a diagram showing the four angles of projection and show 

how they are numbered. 

i 




Fig. 3G. 




Fig. ZH. 




Fig. 37. 



33. How is the revolution of the planes accompHshed so as to bring 

them into coincidence? 

34. How is the observer located in making first angle projections? 



ORTHOGRAPHIC PROJECTION 



43 



35. How is the observer located in making second angle projections? 

36. How is the observer located in making third angle projections? 

37. How is the observer located in making fourth angle projections? 

38. Why are only the first and third angles of projection used in 

drawing? 

39. How does the third angle differ from the first angle projection? 

Show by sketch. 

40. Why is the third angle to be preferred? 




Fig. 3J. 

41. Why is it more difficult to read orthographic projections than oblique 

projections? 

42. What distinct advantages has orthographic projection over oblique 

projection? 





Fig. 3A'. 



Fig. 3L. 



43. Make a complete working drawing of 3-A and show one view in 

section. Assume suitable dimensions. 

Note: A working drawing is a drawing, completely dimensioned, 
with all necessary views for construction purposes. 

44. Make a complete working drawing of 3-B and show one view in 

section. Assume suitable dimensions. 

45. Make a complete working drawing of 3-C and show one view in 

section. Assume suitable dimensions. 

46. Make a complete working drawing of 3-D and show one view in 

section. Assume suitable dimensions. 



44 PARALLEL PROJECTING-LINE DRAWING 

47. Make a complete working drawing of 3-E and show the profile on 

the right, to the to]) view. Assume suitable dimensions. 

48. Make a complete working drawing of 3-F and show the profile on 

the left, to the top view. Assume suitable dimensions. 

49. Make a complete working drawing of 3-G and show the lower view 

as a first angle projection. Assume suitable dimensions. 

50. Rearrange 3-G and show three views in third angle projection. 

51. Make a section of 3-H through the web and observe that the web 

is not sectioned. Assume suitable dimensions. 

52. Make a supplementary view of the 45° ell shown in 3-1. Assume 

suitable dimensions. 

53. Make a supplementary view of 3-J. Assume suitable dimensions. 

54. Make three views of 3-K in third angle projection. Assume suitable 

dimensions. 

55. Make three views of 3-L in third angle projection. Assume suitable 

dimensions. 



CHAPTER IV 
AXONOMETRIC PROJECTION 



401. Nature of isometric projections. Consider three lines 
intersecting at a point and make the angles between each pair 
of lines equal to 120°; the three angles will then total 360° which 
is the total angle about a point. If on one of these lines, or lines 
parallel thereto, lengths are laid off, on the other line, or lines 
parallel thereto, breadths are laid off, and on the remaining line, 
or lines parallel thereto, thicknesses are laid off, it seems quite 
reasonable that a method may be devised whereby the three 
principal dimensions can be plotted so as to represent objects 
in a single view. This method when carried out to completion 
results in an isometric projection. 

Let, in Fig. 34, OA, OB, and OC be the three lines, drawn 
as directed, so that the angles between them are 120°. Suppose 
it is desired to draw a box 

4''X6"X12", made of wood d 

i'' thick. Layoff 12" on OB 
(to any convenient scale), 
6'' on OA and 4" on OC. 
From A, draw a line AD 
parallel to OB and AE par- 
allel to OC; also, from C 
draw CE, parallel to OA and 
CF, parallel to OB. The 
thickness of the wood is to 
be added and the direction 
in which this is laid off is Fig. 34. 

indicated by the direction 

of the corresponding dimension line. In the drawing, the line 
OC was purposely chosen vertical so that OA and OB may be 
readily drawn with a 60° triangle. The other lines, added to 
indicate the construction, are self-explanatory. 

45 




46 



PARALLEL PROJECTING-LIXE DRAWING 



This, then, is an isometric projection, and, as may be noted, 
is a rapid method of representing objects in a single view. Com- 
parison may be made with Figs. 1 and 20, which show the same 
box in obhque and orthographic projection, respectively. 

402. Theory of isometric projection. Let Fig. 35, represent 
a cube shown on the left side of a transparent plane W. If 
the observer, located on the right, projects this cube ortho- 
graphically (projecting lines perpendicular to the plane) on the 
plane W, and at the same time has the cube turned so that each 
of the three visible faces is projected equally on the plane, the 




resultant projection Is an isometric projection of the cube on 
the plane W. 

The illustration on the right of Fig. 35 shows how this cube 
appears when orthographically projected. OA, OB, and OC 
are called the isometric axes. As each face of the cube is initially 
equal to the other faces, and as each edge is also equal, then, 
with equal inclination of the three faces, their projections are 
equal. Hence, the three angles are each equal to 120° and the 
three isometric axes are equal in length. To use these axes 
for drawing purposes merely requires that all dimensions of 
one kind (lengths, for instance) be laid off on any one line, or 
lines parallel thereto, and that this process be observed for the 
three principal dimensions. 

Isometric projection, is, therefore, a special case of ortho- 
graphic projection; because, in its conception, the principal planes 



AXONOMETRIC PROJECTION 



47 



of the object are inclined to the plane of projection. As sohds 
are thereby represented in a single view, only one projection 
is necessary. 

403. Isometric projection and isometric drawing. When 
a line is inclined to the plane of projection, the orthographic 
projection of the line is shorter than that of the line itself, for, if 
the degree of inclination continue, the line will eventually become 
perpendicular to the plane and will then be projected as a point. 
Thus, in Fig. 35, OA, OB, and OC are drawn shorter than the 
actual edges of the cube, and the projection on the right is a true 
isometric projection. In the application of this mode of pro- 




FlG. 



jection, however, it is easier to lay off the actual distance (oi 
any proportion of it) rather than this foreshortened distance. 
If commercial scales are used in laying out the drawing instead 
of the true isometric projections, then it is called an Isometric 
Drawing. 

The distinction between the two is a very fine one, since, 
if the ratio of foreshortening * be used as a scale to which the 
drawing is made, then it is possible by a simple statement, to 
change from isometric drawing to isometric projection. The 
commercial name will be followed and they will bo called isometric 
drawings, always bearing in mind that the distinction nu^ms 
little. 

* This ratio of the actual dimension to the true projected dimension is 
100 : 83 and may be computed by trigonometry. 



48 



PARALLEL PROJECTING-LINE DRAWING 



404. Direction of axes. It is usual to assume one of the 
axes as either horizontal, or vertical, as under these circum- 
stances, the other two can be dra\\Ti with the 60° triangles which 
are standard apphances in the dra\\dng room. Fig. 36 shows 
a wood block in several positions, each of. which is an isometric 
draT\dng; here, the location of the observer with respect to the 
block is at once apparent. 

405. Isometric projection of circles. As no line is shown 
in its true length when it Hes in the faces of the cube and is 
projected isometrically, then, also, no curve that hes in these 
faces can be shown in its true length and, therefore, in its true 
shape. This is due to the foreshortening caused by the inclina- 




FiG. 31 



tion of the plane of the three principal dimensions to the plane 
of projection. 

Fig. 37 shows two cubes, the one on the left appears as a single 
square because it is an orthographic projection and this plane 
was parallel to the plane of projection. On the right, an iso- 
metric drawing is shoTSTi. In the orthographic projection on 
the left, a circle is inscribed in the face of the cube. The circle, 
so drawn, is tangent to the sides of the square at points midway 
between the extremities of the lines. When this square and its 
inscribed circle is sho^Ti isometrically, the points of tangency 
do not change, but, as the square is projected as a rhombus, 
the circle is projected as an ellipse, and is a smooth curve that 
is tangent at mid-points of the sides of the rhombus. A rapid, 



AXONOMETRIC PROJECTION 



49 



though approximate method of drawing an ellipse * is shown 
in the upper face of the isometric cube. The major and the 
minor axes of the elKpse can be laid off accurately by drawing 
the diagonals eg and fh (the notation being alike in both views) ; 
a tangent to the circle is perpendicular to the radius, and, for 
points on the diagonals, this tangent is shown as mn (all four 
are alike because they are equal). Showing mn isometrically 
means that the gm = gn in one view is equal to the gm = gn in 
the other view; the hin = hn in one view is equal to the hni = hn 
in the other view, and so on, for the four possible tangents at 
the diagonals. 

406. Isometric projection of inclined lines and angles. 

Suppose it is desired to locate the centre of a hole in one of the 

faces of a cube as at h (Fig. 38). 

The hole is 12'' back from the 

point f, on the line fm, and 6" up 

from the point m; hfm is the 

isometric representation of a right 

triangle whose legs are 6" and 

12". It will be observed that 

none of the right angles of the 

cube are shown as such, therefore, 

the angle hfm is not the true angle 

corresponding to these dimensions; 

and, hence, it cannot be measured 

with a protractor in the ordinary 

way. 

The point k is located in a similar manner on the top of the 
cube, while ank is the isometric drawing of a right triangle whose 
legs are 9" and 13". A diagonal ed of the cube is also shown. 

The general method of drawing any line in space is to plot the 
three components of the line. For instance, the point d is 
located with respect to the point e, by first laying off ef, then 
fg, and finally gd. 

407. Isometric graduation of a circle. t If tlu^ plane of 
a circki is parallel to the plan(^ of projection, the projection is 

* The exact method of constructing an ellipse will be found in any text- 
book on geometry. 

t This method is also applicable to oblique projection. 




Fig. 38. 



b{) 



PAKALLEL PEOJECTING-LINE DRAWING 



equal to the circle itself and, hence, it can be drawn with a compass 
to the required radius. Any angle is then shown in its true 
size and thus the protractor can be applied in its graduation. 

When circles are shown isometricall}^, however, their pro- 
jections are ellipses, and the protractor graduation is applicable 
no longer. In the upper face of the cube, shown isometrically 
in Fig. 39, the circle is shown as an ellipse whose major and 
minor axes are respectively horizontal and vertical. If, on the 
major axis ab, a semicircle is drawn, and graduated by laying 




Fig. 39. 



off angles at 30° intervals, the points CDEF and G are obtained 
making six angles at 30°, or a total of 180°, the angular measure 
of a semicircle. If, then, the plane of the circle is rotated about 
ab as an axis, until the point E coincides with e, each point of 
division on the semicircle will find itself on the similarly lettered 
point of the ellipse; because, then, the plane of the semicircle 
coincides with the plane of the upper face of the cube. Therefore, 
aoc, cod, doe, etc., are 30° angles, sho^^ni isometrically. 

In the side face of the same cube are shown two methods of 
laying off 45° angles. The fact that both methods locate the 



AXONOMETRIC PROJECTION 



51 



same points tends to show that either method, alone, is applicable. 
The lettering is such that the construction should be clear without 
further explanation. 

408. Examples of isometric drawing. As an example, the 
steps in the drawing of a wooden horse, shown in Fig. 40 and 
used in the building trades for supporting platforms and the 
like, will be followed. The making and the subsequent interpre- 




FiG. 40. 



tation of drawings of this kind is facilitated by the introduction 
of bounding figures of simple shape. In the example chosen, 
the horse is bounded by a rectangular prism. The attached 
dimensions show the necessary slopes and may be introduced 
so as to replace the bounding figure. There is nothing now in 
this drawing and tluTeforo the description will not be needlessly 
exhaustive. The bounding figure, when appended to a drawing 
like that of Fig. 40, helps to emphasize the slopes of the various 
lines. In general, the bounding figure should be removed on 
completion of the drawing. 



52 



PARALLEL PROJECTING-LINE DRAWING 






Fig. 41 shows a toothed wheel. The plane of the circles 
corresponds to the plane of the top face of a cube. To construct 
it, it is necessary to lay out the prism first and then to insert the 
ellipses. The graduation of the circle is similar to that indicated 
in Art. 407. 




Fig. 4L 

A lever is shown in Fig. 42. The centre line ab lies 
in the top face of the lever proper. Circumscribing prisms 
determine the ends of the lever and also the projecting cylin- 
drical ends. 




Fig. 42. 



A gas-engine fly-wheeP is illustrated in Fig. 43. Here the 
fly-wheel is shown so that the planes of the circles correspond 
to one of the side faces of a cube. The wheel has a solid web 
(without spokes) and a quarter of it is removed and shown in 
section. 



AXONOMETRIC PROJECTION 



53 




Ki(.. II 



64 



PARALLEL PEOJECTIXG-LINE DRAWING 



Fig. 44 represents a bell-crank. Again the scheme of using 
base-lines in connection ^^ith circumscribing prisms is sho'^Ti. 
This view should be compared mth that given in Fig. 17. 

409. Dimetric projection and 
dimetric drawing. Let Fig. 45 be 
the isometric projection of a cube, 
with the invisible edges shoTvn dotted. 
A disturbing s^Tiimetry of the lines 
is at once apparent. This objection 
becomes serious when applied to 
drawing if the objects are cubical, or 
nearly so. 

The foregoing difficulty may be 

partially overcome by turning the 

cube so that only two faces are 

projected equally and the remaining 

face may be larger or smaller at pleasure. Fig. 46 shows 

this condition represented. The angle aoc is larger than either 

the angles aob or cob, the latter two (aob and cob) being 




Fig. 45. 




equal. The faces A and B are projected equally, whereas C 
is smaller in this particular case, though it need not be. The 
illustration as shoTsm on the right of Fig. 46 is a dimetric pro- 
jection and, as such, the same scale is applied to the axes oa and 



AXONOMETRIC PROJECTION 



55 



oc because they are projected equally. The axis ob is longer, 
since the corresponding edge is more nearly parallel to the plane 
of projection W than either oa or oc. Hence, to be theoretically 
correct, a different scale must be used on the axis ob. 

When dimetric projection is to be commercially applied, 
confusion may result from the use of two distinct scales. If 
one scale is used on all three axes the combination becomes a 
dimetric drawing. Hence, a dimetric drawing differs from an 
isometric drawing, in so far, as two of three angles are equal 
to each other for the dimetric drawing; whereas, in isometric 
drawing, all three are equal, that is, the axes are 120° 
apart. 

What is true of the direction of the axes in isometric drawing 
is equally true here, that is, the angular relation between the 
axes, alone, determines the type of projection, the direction of 
any one is entirel}^ arbitrary. 

Dimetric drawings do not entirely remove the objectionable 
symmetry, yet they find some use in practice, although they 
present no distinct advantage over any other type. 

410. Trimetric projection and trimetric drawing. Fig. 
47 shows a cube which is held in such a position that the 
orthographic projection of it will 
result in the unequal projection of 
the three visible faces. This, then, 
becomes a trimetric projection. 

To make true projections, three 
different scales must be used. This, 
of course, is objectionable and, hence, 
recourse is had to a trimetric draw- 
ing. A trimetric drawing, therefore, 
requires three axes. The angles be- 
tween the axes differ, but no one 

angle can be a right angle in any case.* The same scale is 
applied to all three. Also, the axes may have any direction, 

* This evidently makes it an oblique ])rojertion. It is impossible to 
project a cube orthograjjliically to j^roduce this; since, if two axes Jire parallel 
to the plane of projection, the third axis must be perpendicular .and is luMiea 
projected as a point. The oblique projection of a cube, which is turned aa 
it would be in isometric drawiiif!;, prescMits no n(>w f(>;iture since it results, 
generally sjieaking, in a trimetric projection. 




Fig. 47. 



56 PARALLEL PROJECTING-LINE DRAWING 

so long as attention is paid to the angular relation be- 
tween them. 

If isometric dramngs introduce the disturbing s}TQmetry, 
then the trimetric is to be recommended, unless one of the other 
types of drawing is found to be more suitable. No examples are 
given in this connection, because the application of dimetric and 
trimetric drawings involve no new features. It is onls" to be 
remembered, that artistic taste may dictate the direction of the 
axes, so as to present the best view of the object to be illustrated. 

411. Axonometric projection and axonometric drawing. 

Isometric, dimetric, and trimetric projection form a group which 
may be conveniently styled as axonometric projections. All 
three are the result of the orthographic projection of a cube, 
so that the three principal axes are projected in a manner as 
already indicated. Axonometric projections are therefore a 
special case of orthographic projection, but their advantages 
are sufficiently prominent to warrant separate classification. 

For isometric projection, one scale is used throughout; for 
dimetric projection, two separate scales are used; and for tri- 
metric projection, three distinct scales are used. When applying 
axonometric projections to draT\dng, the same scale is used on 
all axes, and the group then represents a series w^hich may be 
called axonometric drawing. 

The distinction between isometric projection and isometric 
drawing has been pointed out (Art. 403). It becomes more 
prominent, however, in dimetric and trimetric projection. 

412. Commercial application of axonometric projection. 

For objects of simple shape, with few curves, isometric draTxangs 
serve a useful purpose, because they are easil}^ made and are 
easily read by those unfamiliar with. drax\dng in general. When 
curves are frequent and it is desirable to picture objects in a 
single view, oblique projections offer advantages over isometric 
drawings because it may be possible to make the planes of the 
curves parallel to the plane of projection. The curves will then 
be projected as they actually appear. When curves appear 
in many planes, then orthographic projections xvill answer require- 
ments best, but, as already mentioned, their reading is more 
difficult, due to the simultaneous interpretation of two or more 
views. 




AXOXOMETRIC PROJECTIOX 57 

Before dismissal of the subject of projections in general, 
attention is called to Fig. 48 which is a pecuhar application 
of isometric drawing. It is frequently used as an example of 
an optical illusion. By concentrating 
the \ision at the centre of the picture, 
there seems to be a sudden change from 
six to seven cubes, or vice versa, depend- 
ing upon whether the central corner be 
regarded as a projecting or a depressed 
corner. This is due to the fact that all 
of the cubes are sho^Ti of the same size, 
a condition which is contrary to our 
everyday experience. As objects recede 
from the eye, they appear smaller; and 

in isometric projection there is no correction for this. In fact, all 
the projections so far considered, draw on the imagination for 
their interpretation,* and, therefore, they cannot present a per- 
fectly natural appearance. 

413. Classification of projections. All projections having 
parallel projecting lines may be classified according to the 
method by which they are made. This classification furnishes 
a useful surve}^ of the entire subject and also serves to emphasize 
the distinction between the different methods. 

It will be found, on analysis, that if the object be conceived 
in space with its parallel projecting lines, the oblique projections 
result when the plane of projection cuts the projecting lines 
obliquely. When the plane of projection cuts the projecting 
lines at a right angle, then the orthographic series of projections 
arise. If, still further, the projecting lines, coincide with some 
of the principal lines on the object, and the plane of projection 
is at right angles to the projecting lines, then the two-dimension 
orthographic projections result, and these are connnonly called 
mechanical drawings. If the projecting lines do not coincide 
with some of the principal lines on the object, then the axono- 
metric series of projections follow. 

* The i)rc)je('tions that ovcrcomo these objections are known a^ IVr- 
S]:ective Projections. In these, the projecting hues converge to a point at 
whidi tlie observer is supjmsed to be located. Pliolographs are pers})ec(ives 
in a broad sense. 



58 



PARALLEL PROJECTING-LINE DRAWING 



CLASSIFICATION OF PROJECTIONS HAVING PARALLEL 
PROJECTING LINES 



f Oblique. Pro- 
jecting lines 
inclined to 
plane of pro- 
jection but 
parallel to 
each other. 



Projec- 
tions 



On Plane 

Sui'f aces : 



IncUnation of 
projecting 
lines 45° 
^•ith plane of 
projection, 

< IncUnation of 
projecting 
lines gi-eater 
than 45° 
with plane of 
projection. 

f Mechanical 
Drawin g. 

showing two 
principal di- 
mensions on 
a single view. 
Hence, at 
least two 
views needed 



Orthographic. 

Projecting 
lines perpen- 
d i c u 1 a r to 
the plane of 
projection. 



Axonometric. 

projection, 
showing 
three dimen- 
sions in a 
single view. 



All lines projected 
equal in length. 
Hence, same scale 
used on all. 

Lines parallel to 
plane of projection 
dra\\Ti to equal 
lengt h . Lines per- 
pendic ular to 
plane, projected as 
shorter hnes to di- 
minish distortion. 

Principal hnes of 
object parallel or 
perpendicular to 
planes of projec- 
tion. Sometimes 
called O r t h o - 
graphic projec- 
tion. 

Isometric. All 
thi-ee axes of cube 
projected equally, 
hence, axes are 
120° apart and 
equal in length. 
Apphed as Iso- 
metric Drawing. 

Dimetric. Two of 
the three axes 
projected equally, 
hence, only two 
angles equal. Ap- 
plied as Dimetric 
Drawing. 

Trimetric. Axes 
projected unequ- 
ally, hence, all 
three angles differ. 
Apphed as Tri- 
metric Drawing. 



On 
Curved 

Surfaces : 



Not used in Engineering drawing. 



AXONOMETRIC PROJECTION 59 



QUESTIONS ON CHAPTER IV 

1. What are the isometric axes? 

2. How are they obtained? 

3. What is the angular relation between the pairs of isometric axes? 

4. Show why the isometric projection is a special case of orthographic. 

5. What is the distinction between isometric projection and isometric 

drawing? 

6. What direction do the axes have for their convenient application 

to drawing? 

7. How is a circle projected isometrically? 

8. Show the approximate method of drawing the isometric circle. 

9. How are inclined hnes laid off isometricaUy? 

10. How are angles laid off isometrically? 

11. Show that the laying off of an inclined Hne is accompHshed by laying 

off the components of the line. 

12. How is the isometric circle graduated in the top face of the cube? 

13. Show how the graduation is accomplished in one of the side faces 

of the cube. 

14. What is a dimetric projection? 

15. What is a dimetric drawing? 

16. What angular relation exists between the axes of a dimetric pro- 

jection? 

17. What is a trimetric projection? 

18. What is a trimetric drawing? 

19. What angular relation exists between the axes of a trimetric drawing? 

20. Show why the trimetric drawing eliminates the disturbing S3^mmetry 

of an isometric drawing. 

21. Why cannot the angle between one pair of axes be a right angle? 

22. What are axonometric projections? 

23. Why do projections with parallel projecting Hnes draw on the imag- 

ination for their interpretation? 

24. Draw the object of Question 33 in Chapter 2 in isometric drawing. 

Use bounding figure. 

25. Draw a triangular prism in isometric drawing. Use a j^ounding 

figure. 

26. Make an isometric drawing of a hexagonal prism. Use a bounding 

figure. 

27. Make an isometric drawing of a hexagonal pyramid. l"^sc a bounding 

figure. 

28. Make an isometric drawing of 3-A (Question in Chapter 3). 

29. Make an isometric drawing of 3-B. 

30. Make an isometric drawing of 3-C. 

31. Make an isometric drawing of 3-D. 

32. Make an isometric drawing of 3-F. 

33. Make an isometric drawing of 3-0. 

34. Make an isometric drawing of 3-H. 



60 PARALLEL PROJECTING-LINE DRAWING 

35. Make an isometric drawing of 3-1. 

36. Make an isometric drawing of 3-J. 

37. Make an isometric drawing of 3-K. 

38. Make an isometric drawing of 3-L. - ■ > 

39. Make a complete classification of all projections having parallel 

projecting lines. 



PART II 

GEOMETRICAL PROBLEMS IN ORTHOGRAPHIC 
PROJECTION 



CHAPTER V 
REPRESENTATION OF LINES AND POINTS 

501. Introductory. Material objects are bounded by sur- 
faces, which may be plane or curved in any conceivable way. 
The surfaces themselves are limited by lines, the forms of which 
may be straight or curved. Still further, these lines terminate 
in points. Thus, a solid is really made up of surfaces, lines and 
points, in their infinite number of combinations; and these may 
be considered as the mathematical elements that make up the 
solid. The mathematical elements must be considered as concepts 
as they have no material existence and, hence, are purely imagin- 
ative; that is, a surface has no thickness, therefore, it has no 
volume. A line or a point is a still further reduction along this 
line of reasoning. The usefulness of these concepts must be 
admitted, however, in view of the fact that they play such an 
important role in the conception of objects. 

In general, the outline of any object is found by intuitively 
locating certain points, and joining the points by proper lines; 
the lines, when taken in their proper order determine certain 
surfaces, and the space included between them forms the solid 
(the material object) in question. Hence, to view material objects 
analytically, the nature of their mathematical elements nuist be 
known. 

In Part II of this book, the graphical representation of mathe- 
matical concepts engages the attention. Whether the treatment 
of the subject be from the viewpoint of mathematics or of drawing, 
entirely depends upon the ultimate use. In the two clmpters to 

r.i 



62 GEOMETKICAL PROBLEMS IN PROJECTION 

be presented (V and VI), the method of representing lines, 
points and planes, orthographically, will be discussed. So far, 
attention has been directed to the representation of material 
objects, as common every day experience renders such objects 
familiar to us. 

Before proceeding with the subject in all its detail, familiarity 
must be gained with certain fundamental operations on lines, 
points and planes, as wxll as with their graphical representation 
on two assumed planes called the planes of projection. The 
fundamental operations are grouped and studied without apparent 
reference to future application. It is desirable to do this so as 
to avoid frequent interruptions in the chain of reasoning when 
applying the operations to the solution of problems. 

The student will save considerable time if he is well versed 
in the fundamental operations. In view of this, many questions 
are given at the end of the chapters so that his grasp of the sub- 
ject may be tested from time to time. It may be needless to say 
that the solution of subsequent problems is utterly impossible 
without this thorough grounding. Frequent sketches should 
be made, representing lines, points and planes in positions other 
than shown in this book. These sketches should be made both 
in orthographic and in oblique projection. By this means, the 
student will increase his experience in the subject, much more 
than is possible by all the reading he might do. The proof of 
one's ability always lies in the correct execution of the ideas 
presented. The subject under consideration is a graphical one, 
and, as such, dra^dng forms the test mentioned. It has been 
considered necessary to caution the student so as to avoid the 
complications that will result later, due to insufficient preparation. 
Only in this way will the subject become of interest, to say nothing 
of its importance in subsequent commercial applications. 

502. Representation of the line. Let Fig. 49 be an obhque 
projection of two planes intersecting at right angles to each 
other. The plane W is called the vertical plane of projection 
and HH is called the horizontal plane of projection; these planes 
intersect in a Une XY which is termed the ground line. The 
two planes taken as a whole are knowTi as the principal planes; 
and by their intersection, they form four angles, numbered as 
shown in the figure. In what immediately follows, attention 



KEPRESENTATION OF LINES AND POINTS 



63 



will be concentrated on operations in the first angle of projection, 
and later, extended to all four angles. 

Suppose a cube ABCDEFGH is located in the first angle of 
projection, so that one face Ues wholly in the horizontal plane 
and another face lies wholly in the vertical plane. The two faces 
then lying in the principal planes, intersect in a line which coin- 
cides with the ground line for the conditions assumed. Suppose, 
further, that it is ultimately desired orthographically to represent 
the diagonal AG of the cube. For the present, the reasoning 
will be carried out by the aid of the oblique projection. 

The construction of the horizontal projection consists in drop- 
ping upon the horizontal plane, perpendiculars from points on 




Fig. 49. 



the line. For the line AG, the projecting perpendicular from 
the point A is AE, and E is then the horizontal projection of the 
point A in space.* As for the point G on the line, that already 
lies in the horizontal plane and is its own horizontal projection. 
Thus, two horizontal projections of two points on the line are 
established, and, hence, the horizontal projection of the line is 
determined by joining these two projections. This is true, 
because all the perpendiculars from the various points on the line 
lie in a plane, which is virtually the horizontal projecting plane 
of the line. It cuts the horizontal plane of projection in a line 
EG, which is the horizontal projection of the line AG in space. 

* It is to be noted that the i)r()jo('lion of a point is found at the place 
where the projecting hnc pierces the plane of projection. 



64 



GEOMETRICAL PROBLEMS IX PROJECTIONS 



Putting this in another form, the projection EG gives the san.j 
mental impression to an abserver viewing the horizontal plane 
from above as does the line AG itself; in fact, EG is a drawing 
of the line AG in space. 

Directing attention for a moment to the vertical plane, it is 
found that the construction of the vertical projection consists 
in dropping a series of perpendiculars from the line AG to that 
plane. For point G, the perpendicular to the vertical plane is 
GH, and H is thus the vertical projection of G. The point A 
lies in the vertical plane and, hence, is its own vertical projection. 
A line joining A and H is the vertical projection of AG. Again, 
a plane passed through AG, perpendicular to the vertical plane, 




Fig. 49. 



will cut from it the line AH, which is the vertical projection as 
has been determined. Thus, AH is a drawing of AG because it 
conveys the same mental impression to an observer who views 
it in the way the projection was made. 

503. Line fixed in space by its projections. The location 

of the principal planes is entirely arbitrary, as is also the line in 
question; but, when both are once assumed, the line is fixed in. 
space by its projections on the principal planes. If a plane be 
passed through the horizontal projection EG (Fig. 49), perpen- 
dicular to the horizontal plane, it will contain the line AG in. 
space, since the method is just the reverse of that employed in 
finding the projection. Similarly, a plane through the vertical 
projection, perpendicular to the vertical plane, will also contain 



REPRESENTATION OF LINES AND POINTS 



65 



the line AG. It naturally follows that the line AG is the inter- 
section of the horizontal and vertical projecting planes and, 
therefore, the line is absolutely fixed, with reference to the prin- 
cipal planes, by its projections on those planes. 

504. Orthographic representation of a line. The line that 
has been considered so far is again represented in the left-hand 
view of Fig. 50, in obhque projection. The edges of the cube 
have been omitted here in order to concentrate attention to the 
line in question. AG is that line, as before, EG is its horizontal 
projection and AH is its vertical projection.* 

Suppose that the plane HH is revolved in the direction of the 
arrows, 90° from its present position, until it coincides with the 




Fig. 50. 



vertical plane W. The view on the right of Fig. 50 shows the 
resultant state of affairs in orthographic projection. AH is the 
vjertical projection and EG is the horizontal projection. AE in 
one view is the equivalent of AE in the other; HG and EH in 
one view are the equivalents of HG and EH in the other. Both 
views represent the same line AG in space. At first sight, it may 
appear that the oblique projection is sufficiently clear, and such 
is the case; but, in the solution of problems, the orthograi)hic 
projection as shown on the right possesses many advantages. In 
due time, this mode of representation will be considered, alone, 
without the oblique projection. 

* The projections of a line may bo considered ;us sliadows on their resi)ective 
pianos. Tlio lipht comes in parallel rays, pcMpondicularly drreclod to Iho 
planes of i)rojection. See also Arts. '2i)'.l and iUO in this eoimeotion. 



66 



GEOMETRICAL PROBLEMS IN PROJECTION 



505. Transfer of diagrams from orthographic to oblique 
projection. It is desirable to know how to transfer diagrams 
from one kind of projection to the other. If the orthographic 
projection appears confusing, the transfer to oblique projection 
may be of service. On the right of Fig. 50, is given the ortho- 
graphic projection of the line AG. To construct the oblique 
projection from this, draw the principal planes as shown in the 
left-hand diagram. From any point E on the ground line in the 
oblique projection, lay off EA, vertically, equal to EA in the 
orthographic projection. On the sloping line (ground line), 
lay off EH equal to EH in the orthographic projection. Then 
draw HG in the horizontal plane, equal to HG in the orthographic 
projection. A line joining A and G in the oblique projection 




Fig. 50. 

gives the actual line in space; hence, in the oblique projection, 
the actual line and both of its projections are shown. In the 
orthographic projection, only the projections are given; the 
line itself must be imagined. Compare the method of constructing 
the oblique projection with Art. 207 and note the similarity. 



506. Piercing points of lines on the principal planes. 

If AG (Fig. 50) be considered as a Umited portion of line indefi- 
nitely extended in both directions, then A and G are the piercing 
points on the vertical and horizontal planes respectively. On 
observation, it will be found that the vertical piercing point lies 
on the vertical projection of the fine, and also on a perpendicular 
to the groijnd line XY from the point where the horizontal pro- 
jection intersects the ground line; hence, it is at their intersection. 



REPRESENTATION OF LINES AND POINTS 



67 



Likewise, the horizontal piercing point Ues on the horizontal 
projection of the line and also on the perpendicular erected at 
the intersection of the vertical projection with the ground line; 
hence, again, it is at the intersection of these two lines. 

In Fig. 51 let ab be the horizontal projection of a line AB in 
space, and a'b' be the corresponding vertical projection. The 
line, if extended, would pierce the horizontal plane at c and the 
vertical plane at d'. This hne is shown as an obhque projection 
in Fig. 52. All the hnes that are required in the mental process 
are shown in this latter view. The actual construction in ortho- 
graphic projection is given in Fig. 51. To locate the vertical 
piercing point prolong both projections, and at the point where 
the horizontal projection intersects the ground line, erect a per- 





FiG. 51. 



Fig. 52. 



pendicular until it intersects the prolongation of the vertical 
projection. This locates the vertical piercing point. To find 
the horizontal piercing point, prolong the vertical projection 
until it intersects the ground line, and at this point, erect a per- 
pendicular to the ground line. Then, the point at which this 
perpendicular intersects the prolongation of the horizontal pro- 
pection is the horizontal piercing point. 

A convenient way of looking at cases of this kind is to assume 
that XY is the edge of the horizontal plane while viewing the vert i- 
cal projection, therefore, AB nuist pierce the horizontal ]ilane 
somewhere in a line perpendicular to the vertical plane at the 
point c'. In viewing the horizontal jilane, XY now represents 
the vertical plane on edge. H(Te, again, Ww Vmv AB must pierce 



68 



GEOMETRICAL PROBLEMS LS' PROJECTION 



the vertical plane somewhere in 
zontal plane at the point d. 



line perpendicular to the hori- 



507. Nomenclature of projections. In what follows, the 
actual object will ;r :.T-:^:i::.:r:i ::y :-e capital letters, as the line 
AB for instance. The horizontal projecfions will be indicated 
by the small letters as the line ab, and the vertical projections 
by the smaU prime letters as the line ab . It should always be 
remembered that in orthographic projection, the projections 
alone are given, the actual object is to be imagined. 

508. Representation of points. A single point in space is 
located with respect to the principal planes as shown 'm. Fig. 53. 
A is the actual point while a is its horizontal, and a' its vertical 
projection. The distance of A above the horizontal plane is equal 
to the length of its projecting perpendicular Aa and this is equal 
to a'o because Aa and a'o are both perpendicular to the hori- 




FiG. .53. 



zontal plune HH. Also. Aa' and ao are perpendicular to the verti- 
cal plane, tnerdore. the figure Aaoa' is a rectangle, whose opjxxsite 
sides are necessarily equal. Hence, also, the distance of A from 
the vertical plane is equal to the length of its projecting perpen- 
dicular Aa' which also equals ao. Performing the usual revolu- 
tion of the horizontal plane, a wiU reach a" on a line a'a" which 
is perpendicular to the ground line XY. It has been shown that 
a'o is perpendicular to XY and it only remains to prove that oa'' 
is a continuation of a'o. This must be so, because a revolves 
about the ground line as an axis, in a plane determined by the 
two intersecting lines a'o and ao. This plane cuts from the ver- 



KEPRESENTATION OF LINES AND POINTS 69 

tical plane the line a'a" which is perpendicular to XY because a 
portion (a'o) of it is perpendicular. As the point a revolves 
about XY as an axis, it describes a circle, whose radius is oa, 
and hence oa" must equal oa. 

In projection,* this is shown on the right-hand diagram of 
Fig. 53. Both figures are lettered to correspond as far as con- 
sistent. The actual point A is omitted, however, in the right- 
hand diagram because the very object of this scheme of repre- 
sentation is to locate the point from two arbitrary planes (prin- 
cipal planes), solely by their projections on those planes. This 
latter is an exceedingly important fact and should always be 
borne in mind. 

509. Points lying in the principal planes. If a point 
lies in one of the principal planes, it is its own projection in that 
plane and its corresponding projection 

lies in the ground hne. Fig. 54 shows T* 

such cases in projection. A is a point j^ g^ c^p' 

lying in the vertical plane, at a distance j 

a'a above the horizontal plane; its ver- ij 

tical projection is a' and its correspond- Yig. 54. 

ing horizontal projection is a. B is 

another point, lying in the horizontal plane, at a distance bb' 
from the vertical plane; b is its horizontal and b' the correspond- 
ing vertical projection. 

If a point lies in both planes, the point coincides with both 
of its projections and must therefore be in the ground line. C is 
such a point, and its two projections are indicated by cc', both 
letters being affixed to the one point. The use of these cases 
will appear as the subject develops. 

510. Mechanical representation of the principal planes. 

For the time being, the reader may find it desirable to construct 
two planes f so that lines and points may be actually represented 

* Hereafter, orthographic projection will usually be designated simply 
as "in projection." 

t For chissroom work, a more serviceable device can be made of hinged 
screens, constructed of a fine mesh wire. Wires can be ejisily insert (h1 to 
represent lines and tlu^ yn-ojections drawn with chalk. The revolution of 
th(i planes can be accomplished by properly hinging the planes so that they 
can be made to lie approximately Hat. 



70 



GEOMETRICAL PROBLEMS IX PROJECTION 



Fig. 55. 



with reference to them. If two cards be sht as shown in Fig. 55, 
they can be put together so as to represent two planes at right 

angles to each other. Lines may be 
represented by use of match-sticks and 
points by pin-heads, the pin being so 
inserted as to represent its projecting 
perpendicular to one plane. The idea 
is recommended until the student be- 
comes familiar wdth the involved opera- 
tions. As soon as this familiarity is obtained, the cards should 
be dispensed with, and the operations reasoned out [in space, as 
far as possible, without the use of any diagrams. 

511. Lines parallel to the planes of projection. Assume 
a line parallel to the horizontal plane. Evidently this line cannot 
pierce the horizontal plane on account of the parallelism. It 
will pierce the vertical plane, however, if it is inclined to that plane . 




Fig. 56. 



In Fig. 56 this line is shoT\Ti both in oblique and orthographic 
projection. The piercing point on the vertical plane is found 
by erecting a perpendicular at the point where the horizontal 
projection intersects the ground Hne. It will also He on the ver- 
tical projection of the line. As the line is parallel to the horizontal 
plane, its vertical projection is parallel to the groimd Une. The 
piercing point ^ill therefore lie at the intersection of the two 
lines; that is, of the perpendicular from the ground line and the 
vertical projection of the Une. In both views of the figure, AB 
is the given line and b' is its piercing point. 

If, on the other hand, the line is parallel to the vertical plane 



REPRESENTATION OF LINES AND POINTS 



71 



(Fig. 57) and inclined to the horizontal plane, it v/ill pierce the 
horizontal plane at some one point. Its horizontal projection 
is now parallel to the ground line XY. The horizontal piercing 




Fig. 57. 



point is at b, as shown, and is found in much the same way as that 
in the illustration immediately preceding. 

A case where the Hne is parallel to both planes is shown in 




X-4- 



-t-Y 



Fia. 58. 



Fig. 58. This line cannot pierce either plane and, therefore, both 
its projections must be parallel to the ground line as depicted. 

512. Lines lying in the planes of projection. If a line 

lies in the plane of projection, it is 

its own projection in that plane, and 

its corresponding projection lies in 

the ground line. When the line lies 

in both planes of projection it must 

therefore coincide with the ground line. 

Fig. 59 shows in projection the three 

., 1 r,M n ^ • ,. Fig. 59. 

cases possible. 1 he first is a line 

lying in the horizontal plane, ab is its horizontal projection nml 




72 



GEOMETEICAL PEOBLEMS IN PROJECTION 



d' 



x-4- 



ff 
-> — 



Fig. 59. 



a'b' is its corresponding vertical projection. The second is a 

line lying in the vertical plane, c'd' 
is its vertical projection and cd is its 
corresponding horizontal projection. 
The third case shows a line in both 
planes and its horizontal and vertical 
projections coincide with the ground 
line and also the line itself; the 
coincident projections are indicated 

as shown at ee' and ff , read ef and e't'. 

513. Lines perpendicular to the planes of projection. 

If a line is perpendicular to the horizontal plane, its projection 
on that plane is a point, because both projecting perpendiculars 
from the extremities must coincide with the perpendicular line. 
The vertical projection, however, shows the line in its true length, 
perpendicular to the ground line, as such a line is parallel to the 



kab 



c'd' 

T 
I 



>^ 



D^ 



-Y H- 



FiG. 60. 



lA 



vertical plane. Fig. 60 shows this in projection as AB, and, in 
addition a profile plane is added which indicates the fact more 
clearly. CD is a line perpendicular to the vertical plane with one 
extremity of the line in that plane. In the profile plane, the 
lines AB and CD appear to intersect, but this is not necessarily 
the case. The construction of the projections of the line CD is 
identical with that immediately preceding. 

514. Lines in all angles. So far, the discussion of lines 
and points in space has been limited entirely to the first angle. 
If a line is indefinite in extent, it may pass through several of the 
four angles. A case in each angle will be taken and the salient 
features of its projections will be pointed out. 

Fig. 61 shows a line passing through the first angle. It con- 
tinues beyond into the second and fourth angles. In projection. 



REPRESENTATION OF LINES AND POINTS 



73 



the condition is shown on the right. The projections ab and a'b' 
show those of the Hmited position that traverses the first angle. 




The dotted extensions show the projection of the indefinite line, 
and are continued, at pleasure, to any extent. 

A line in the second angle is shown in Fig. 62. The horizontal 
projection is ab and the ver- 
tical projection is a'b'. If 
the horizontal plane be re- 
volved into coincidence with 
the vertical plane, the view 
in projection will show that, 
in this particular case, the 

projections of the lines cross Pj^ g2. 

each other. Only the limited 

portion in the second angle is shown, although the line may be 
indefinitely extended in both directions. 

A third angle line is illustrated in Fig. 63. It may be observed, 





Fig. 63. 

in comparison with a line in the first angle, that the horizontal 
and vertical projections are interchani2;ed for the liniitinl portion 



74 



GEOMETRICAL PROBLEMS IN PROJECTION 



of the line shown. In other words the horizontal projection 
of the Une is above the ground Une and the vertical projection 
is below it. The case may be contrasted with Fig. 61. 

The last case is shown in the fourth angle and Fig. 64 depicts 




Fig. 64. 

this condition. Both projections now cross each other below 
the ground line. Contrast this with Fig. 62. The similarity 
of the foregoing and Arts. 315, 316 and 317 may be noted. 

515. Lines with coincident projections. Lines, both of 
whose projections lie in the ground Une have been previously 
considered (512). If a line passes through the ground line from 




Fig. 65. 



the second to the fourth angle, so that anj^ point on the line is 
equidistant from both planes, then the two projections of the 
line will be coincident. Fig. 65 illustrates such a condition. 
The hne is not indeterminate by having coincident projections, 
however, because the oblique projections can be constructed 



REPRESENTATION OF LINES AND POINTS 



iO 



from the orthographic representation. The projection on the 
profile plane shows that this line bisects the second and fourth 
angles. 

If the line passes through the ground line, from the second 
to the fourth angles, but is not equidistant from the principal 
planes, then both projections will pass through the same point 
on the ground line. 

516. Points in all angles. Fig. 66 shows four points A, B, C, 
and D, lying, one in each angle. The necessary construction 
lines are shown. On the right, the condition is depicted in 
projection, the number close to the projection indicating the angle 
in which the point is located. 

Observation will show that the first and third angle projec- 




X- 



14 

I 

id 
id' 



Fig. 66. 



tions are similar in general appearance, but with projections 
interchanged. The same is true of the second and fourth angles; 
although, in the latter cases, both projections fall to one or the 
other side of the ground line. 

517. Points with coincident projections. It may be 

further observed in Fig. 66 that in the second and fourth angles, 
the projection b, b' and d, d' may be coincident. This simply 
means that the points are equidistant from the planes of projection. 
The case of the point in the ground line has been noted 
(508), such points lie in no particular angle, unless a new set of 
principal planes be introduced. It can then be considered under 
any case at pleasure, depending upon the location of the })rin- 
cipal planes. 

518. Lines in profile planes. A line may be located so tiiat 
both of its projections are perpenilicular to the ground line. 



76 



GEOMETRICAL PROBLEMS IN PROJECTION 



The projections must therefore be coincident,* since they pass 
through the same point on the ground hne. Fig. 67 shows an 
example of this kind. Although the actual line in space can be 
determined from its horizontal projection ab and its vertical 




Fig. 67. 

projection a'b', still this is only true because a limited portion 
of the line was chosen for the projections. The profile shown on 
the extreme right clearly indicates the condition. The location 
of the profile with respect to the projection should also be noted. 
The view, is that obtained by looking from right to left, and is 
therefore located on the right side of the projection. The number- 
ing on the angles should also help the interpretation. 




A H 



h>. 



I 



9nk' 
'a' 



■Y, 



Fig. 68. 



Fig. 68 shows a profile (on the left-hand diagram) of one line 
in each angle. The diagram on the right shows the lines in 
projection. The numbers indicate the angle in which the line 
is located. 



* Compare the coincident projections in this case with those of Art. 515. 



REPRESENTATION OF LINES AND POINTS 77 



QUESTIONS ON CHAPTER V 

1. Discuss the point, the hne, and the surface, and show how the material 

object is made up of them. 

2. What are the mathematical elements of a material object? 

3. Why are the mathematical elements considered as concepts? 

4. How is the outline of a material object determined? 

5. What is meant by the graphical representation of mathematical con- 

cepts? 

6. What are the principal planes of projection? 

7. What is the ground line? 

8. How many dihedral angles are formed by the principal planes and 

how are they numbered? Make a diagram. 

9. How is a line orthographically projected on the principal planes? 

10. How many projections are required to fix the hne with reference to 

the principal planes? Why? 

11. Show how one of the principal planes is revolved so as to represent a 

line in orthographic projection. 

12. Assume a line in the first angle in orthographic projection and show 

how the transfer is made to an obhque projection. 

13. Under what conditions will a hne pierce a plane of projection? 

14. Assume a line that is inclined to both planes of projection and show 

how the piercing points are determined orthographically. Give 
the reasoning of the operation. 

15. Do the orthographic projections of a line represent the actual line in 

space? Why? 

16. Show a point in oblique projection and also the projecting lines to the 

principal planes. Draw the corresponding diagram in orthographic 
projection showing clearly how one of the principal planes is re- 
volved. 

17. Draw, in projection, a point lying in the horizontal plane; a point 

lying in the vertical plane; a point lying in both planes. Observe 
nomenclature in indicating the points. 

18. Indicate in what angle the points shown ^^ 

in Fig. 5-A are located. 

19. Draw a line parallel to the horizontal 

plane but inclined to the vertical plane 
in orthographic projection. Transfer 
diagram to oblique j)rojoction. 

20. Draw a line parallel to the vertical j^lane 

but inclined to the horizontal ])lane in 
orthographic projection. Transfer dia- 
gram to ()})li(iue projection. 

21. Draw a line parallel to both princi]ial planes in ortliographic i)ro- 

jection. Transfer diagram to oblique projection. 

22. When a line is parallel to both principal planes is it parallel to the 

ground line? Why? 



1 i 




d' 


r 


*6 






1 *^ 


Fig. 


5- 


-A 





78 



GEOMETRICAL PROBLEMS IN PROJECTION 



23. In Fig. 5-B, make the oblique projection of the hne represented. 

24. Draw the projections of a hne Ij^ing in the horizontal plane. 

25. Draw the projections of a line lying in the vertical plane. 

26. Draw the projections of a line lying in the ground hne, observing the 

nomenclature in the representation. 




Fig. 5-B. 



Fig. 5-C. 



27. In Fig. 5-C give the location of the hues represented by the ortho- 

graphic projections. Construct the corresponding obhque projec- 
tions. 

28. Show two lines, one perpendicular to each of the planes and also 

draw a profile plane for each indicating the advantageous use in 
such cases. 

29. When a line is perpendicular to the horizontal plane, how is it pro- 

jected on that plane? 

30. When a line is perpendicular to the vertical plane, why is the horizon- 

tal projection equal to it in length? 

31. Draw a hne in the second angle, in orthographic projection. Transfer 

the diagram to obhque projection. 

32. Draw a line in the third angle, in orthographic projection. Transfer 

the diagram to oblique projection. 

33. Draw a hne in the fourth angle, in orthographic projection. Transfer 

the diagram to obhque projection. 

34. In Fig. 5-D, specify in which angles each of the lines are situated. 




Fig. 5-D. 



35. Draw hnes similar to 5-D in orthographic projection and transfer 

the diagrams to oblique projection. 

36. Make the orthographic projection of a hne with coincident projections 

and show by a profile plane what it means. Take the case where 
the line passes through one point on the ground line and is per- 
pendicular to it, and the other case where it is inclined to the 
ground line through one point. 



REPRESENTATION OF LINES AND POINTS 79 

37. Locate a point in each angle and observe the method of indicating 

them. 

38. Make an obhque projection of the points given in Question 37. 

39. Make an orthographic projection of points with coincident pro- 

jections and show under what conditions they become coincident. 

40. In what angles are coincident projections of points possible? Show 

by profile. 

41. Make the obhque projections of the lines shown in Fig. 68 of the 

text. Use arrows to indicate the lines. 

42. Why is it advantageous to use a profile plane, when the hnes are 

indefinite in extent, and he in the profile plane? 



CHAPTER VI 



REPRESENTATION OF PLANES 

601. Traces of planes parallel to the principal planes. 

Let Fig. 69 represent the two principal planes by HH and W 
intersecting in the ground line XY. Let, also, RR be another 
plane passing through the first and fourth angles and parallel to 
W. The plane RR intersects the horizontal plane HH in a line 
tr, which is called the trace of the plane RR. As RR is parallel 




Fig. 69. 

to W, the case is that of two parallel planes cut by a third plane, 
and, from solid geometry, it is known that their intersections 
are parallel. If the horizontal plane is revolved, by the usual 
method, into coincidence with the vertical plane, the resulting 
diagram as shown on the right will be the orthographic represen- 
tation of the trace of a plane which is parallel to the vertical 
plane. 

If, as in Fig. 70, the plane is parallel to the horizontal plane, 
the condition of two parallel planes cut by a third plane again 
presents itself, the vertical plane of projection now being the 
cutting plane. The trace t'r' is then above the ground line, 
and, as before, parallel to it. 

602. Traces of planes parallel to the ground line. When 
a plane is parallel to the ground line, and inclined to both planes 

80 



REPRESENTATION OF PLANES 



81 



of projection, it must intersect the principal planes. The line 
of intersection on each principal plane will be parallel to the ground 




^ 



B 



i- 



:::£22:^, 




X- 



FiG. 70. 



line because the given plane is parallel to the ground line and, 
hence, cannot intersect it. Fig. 71 shows this condition in oblique 




Fig. 71, 



and orthographic projection, in which tr is the horizontal trace 
and t'r' is the vertical trace. 

A special case of this occurs if the plane passes through the 
ground line. Both traces then coincide with 
the ground line and the orthographic repre- 
sentation becomes indeterminate unless the 
profile plane is attached. 

Fig. 72 is a profile and shows several 
planes passing through the ground line, each 
of which is now determined. It may be 
possible, however, to introduce a new hori- 
zontal plane H'H', parallel to the principal 
horizontal plane. Such an artifice will bring the case into that 
imnuMliately preceding. It may then be shown as an ordinary 



V 
T 

H \ 


'X H 


H' yl 


\ ,t 


"J 


V 



Fig. 72. 



82 



GEOMETRICAL PROBLEMS IN PROJECTION 



orthographic projection, just as though the original horizontal 
plane were absent. The same would still be true if a new ver- 
tical plane were added instead of the horizontal plane, or, if an 
entirely new set of principal planes were chosen so that the 
new principal planes would be parallel to the old ones. 

603. Traces of planes perpendicular to one of the prin» 
cipal planes. Fig. 73 shows a plane that is perpendicular to 




Fig. 73. 



the horizontal plane but inclined to the vertical plane. It may 
be imagined as a door in a wall of a room. The angle with the 
vertical plane (or wall in this case) can be changed at will by 
swinging it about the hinges, yet its plane always remains per- 







\t \t \t 



Fig. 74. 



pendicular to the horizontal plane (floor of the room). The inter- 
section with the vertical plane is perpendicular to the horizontal 
plane because it is the intersection of two planes (the given 
plane and the vertical plane), each of which is perpendicular to 
the horizontal. As a consequence the vertical trace is perpen- 
dicular to the ground line, because, when a line is perpendicular 



REPRESENTATION OF PLANES 



83 



to a plane it is perpendicular to every line through its foot (from 
geometry). The orthographic representation of three distinct 
planes is shown on the right; the cases selected all show Tt' 
perpendicular to the ground line. 

What is true regarding planes perpendicular to the horizontal 
plane is equally true for planes similarly related with respect to the 
vertical plane. Fig. 74 gives an illustration of such a case. Here, 
the horizontal trace is perpendicular to the ground line but the 
vertical trace may make any angle, at will, with the ground line. 

604. Traces of planes perpendicular to both principal 
planes. When a plane is perpendicular to both principal planes, 




Fig. 75. 



its two traces are perpendicular to the ground line. Such a plane 
is a profile plane and is shown in Fig. 75. 

605. Traces of planes inclined to both principal planes. 

It may be inferred from the preceding, that, if the plane is 




Fig. 76. 

inclined to both principal planes, neither trace can be perpen- 
dicular to the ground line. Fig. 76 shows such a case, the 



84 



GEOMETRICAL PROBLEMS IN PROJECTION 



u' 








i /t 


- 



T/:_ 



t.- 



/ 



Fig. 77. 



orthographic representation of which in projection is shown on 
the right. 

Fig. 77 shows two cards, each slit half waj-, as indicated. 
These can be fitted together to represent the principal planes. 

If at any point, a and a', on XY 
slits tt and MM be made, it will 
be found on assembling the 
cards that a third card can be 
inserted. 

In the upper card, another 
slit s's' may be made, through 
the point a', with its direction 
parallel to tt. As in the previous 
instance, a card can again be 
inserted. This case is of in- 
terest, however, because both traces become coincident on the 
revolution of the horizontal plane, and fall as one straight line 
tTt'; as sho^Ti in projection in Fig. 77. 

606. Traces of planes intersecting the ground line. It 

must have been observed in the cases where the given plane is 
inclined to the ground line that both traces pass through the 
same point on the ground line. This becomes further evident 
when it is considered that the ground line can intersect a plane 
at but one point, if at all. This one point lies in the ground line, 
and, hence, it has coincident projections (509). 

607. Plane fixed in space by its traces. Two intersect- 
ing lines determine a plane (from solid geometry). Hence, the 
two traces of a plane fix a plane with reference to the principal 
planes because the traces meet at the same point on the ground 
line. If the diagram is such that the traces do not intersect in 
the ground line within the limits of the drawing, it is assumed 
that they will do so if sufficiently produced. The limiting posi- 
tion of a plane whose traces cannot be made to intersect the 
ground line is evidently a plane parallel to it. This plane is 
then parallel to the ground line and its traces must also be parallel 
to it (602). 

608. Transfer of diagrams from orthographic to 
oblique projection. Let the right-hand diagram of Fig. 



EEPRESENTATION OF PLANES 



85 



78 represent a plane in orthographic projection. Through 
any point S, on the ground hne, pass a profile plane sSs', inter- 
secting the two traces tT and Tt' at points o and p. To transfer 
the orthographic to the oblique projection, lay off the principal 
planes HH and W as shown, intersecting in the ground line 




Fig. 78. 

XY. Lay off any point S on the ground line in the oblique pro- 
jection and then make So and Sp of the orthographic equal to 
the similarly lettered lines in the oblique projection. The profile 
plane sSs' is therefore determined in the oblique projection. 
Make TS of one diagram equal to the TS of the other, and com- 



< X 




T/ 



.)i 



!/ 



>• 



plete by drawing the traces through T, o and T, p. To increase 
the clarity of the diagram, a rectangular ])lano may be shown 
as though it passes through the i)rin(*ipal planes. Fig. 78 has 
this plane added. 

A case when^ the two tracers an* coincidc^nt is shown in Fig. 
79. Two profile planes are recjuired, but only one trace of each 



86 GEOMETRICAL PROBLEMS IN PROJECTION 

is needed. All the necessary construction lines are shown in this 
diagram. 

609. Traces of planes in all angles. As planes are indef- 
inite in extent, so are their traces; and, therefore, the traces are 
not limited to any one angle. In the discussion of most problems, 
it may be possible to choose the principal planes so as to limit 
the discussion to one angle — usually the first. The advantage 
to be gained thereby is the greater clarity of the diagram, as then 
the number of construction hues is reduced to a minimum. Third 
angle projection may also be used, but the transfer to oblique 
projection is undesirable. Second and fourth angles are avoided 
because the projections overlap (317, 514, 516). 

Fig. 80 shows the complete traces of a given plane T. The 



\ 



A 



'^—Y \ ^x y y 

/ ^ \ /^ \ 

/t f\^ t^ 

Fig. 80. Fig. 81. 



full lines indicate the traces in the first angle; the dotted lines 
show the continuation in the remaining three angles. Fig. 81 
represents each quadrant separately for the same plane that is 
shown in Fig. 80. The appended numbers indicate the angle 
to which the given plane is limited. 

610. Projecting plane of lines. It is now evident that 
the finding of the projection of a line is nothing more nor 
less than the finding of the trace of its projecting plane. The 
two perpendiculars from a line to the plane of projection are 
necessarily parallel and therefore determine a plane. This plane 
is the projecting plane of the line and its intersection (or 
trace) with the plane of projection contains the projection of the 
line. 

Manifestly, any number of lines contained in this projecting 
plane would have the same projection on any one of the principal 
planes, and that projection therefore does not fix the line in space. 



REPRESENTATION OF PLANES 



87 



If the projection on the corresponding plane of projection be taken 
into consideration, the projecting planes will intersect, and this 




Fig. 82. 

intersection will be the given line in space. Fig. 82 illustrates 
the point in question. 



QUESTIONS ON CHAPTER VI 

1. What is the trace of a plane? 

2. Draw a plane parallel to the vertical plane passing through the first 

and fourth angles, and show the resulting trace. Make diagram 
in obhque and orthographic projection. 

3. Take the same plane of Question 2 and show it passing through the 

second and third angles. 

4. Show how a plane is represented when it is parallel to the horizontal 

plane and passes through the first and second angles. Make 
diagram in oblique and orthographic projection. 

5. Take the same plane of Question 4 and show the trace of the plane 

when it passes through the third and fourth angles. 

6. Show how a plane is represented when it is parallel to the ground 

fine and passes through the first angle. 

7. How is a plane represented when it is parallel to the ground line and 

passes through the second angle? Third angle? Fourtli Anglo? 

8. Show how a plane passing through the ground lino is indotorminato 

in orthographic projection. 

9. When a plane passes through the ground lino show how tho i^rofilo 

plane might be used to advantage^ in r(^])rosonting tho piano. 

10. When a piano passes through tho ground lino, show how an auxiliary 

principal piano may bo usod to obtain dotorminato traoos. 

11. When a piano is iKMpondicular to tho horizontal piano and inolinod to 

tho vorti(;al piano, show how this is roprosontod orthographioally. 
Make, also, the oblique projection of it. 



88 GEOMETRICAL PROBLEMS IN PROJECTION 

12. When a plane is perpendicular to the vertical plane and inclined to 

the horizontal plane show how this is represented orthographically. 
Make, also, the obhque projection of it. 

13. When a plane is perpendicular to both principal planes draw the 

orthographic traces of it. 

14. Why is the ground hne perpendicular to both traces in Question 13? 

Is this plane a profile plane? 

15. When a plane is inchned to the ground line show how the traces are 

represented. 

16. Why do both traces of a plane intersect the ground hne at a point 

when the plane is inclined to it? 

17. Why do the traces fix the plane with reference to the principal planes? 

\ / / < 

X ^ ^ ? ^-f Y 

Fig. 6-A. 

18. When a plane is parallel to the ground hne, why are the traces of the 

plane parallel to it? 

19. Show an obhque plane in all four angles of projection (use only the 

limited portion in one angle and use the same plane as in the 
illustration). 

20. In what angles are the planes whose traces are shown in Fig. 6-A? 



x-^ 



Fig. 6-B. Fig. 6-C. 

21. Show how a line in space and its projecting perpendiculars determine 

a plane which is the projecting plane of the line. Is the projection 
of the line the trace of the projecting plane of the hne? 

22. Is it possible to have two separate fines whose projections are coin- 

cident on one plane? How are such fines determined? 

23. Given the traces of a plane in orthographic projection as shown in 

Fig. 6-B construct the oblique projection of it. 

24. Construct the obhque projection of a plane having coincident projec- 
tions (Fig. 6-C). 



CHAPTER Vn 
ELEMENTARY CONSIDERATIONS OF LINES AND PLANES 

701. Projection of lines parallel in space. When two 
lines in space are parallel, their projecting planes are parallel, 
and their intersection with any third plane will result in parallel 
lines. If this third plane be a plane of projection, then the 
traces of the two projecting planes will result in parallel projections. 

Fig. 83 shows two Hues, AB and CD in space. The Unes are 





Fig. 83. 



Fig. 84. 



shown by their horizontal projections ab and cd, which are parallel 
to each other, and by their vertical projections a'b' and c'd', 
which are also parallel to each other. A perfectly general case 
is represented pictorially on a single plane of projection in Fig. 84. 



702. Projection of lines intersecting in space. 

lines in space intersect, their projections 
intersect, because the two lines in space must 
meet in a point. Further, the projection of 
this point must be common to the projections j^J\4 
of the lines. 

In Fig. 85 two such lines, AB and CD are 
shown, represented, as usual, by their pro- 
jections. O is the intersection in space, Fig. 85. 
indicated by its horizontal and vertical projections o and 

89 



If two 




^ 



90 



GEOMETRICAL PROBLEMS IN PROJECTION 



respectively. EFand GH (Fig. 86) are two other lines, chosen 
to show how in the horizontal plane of projection, the two pro- 
jections may coincide, because the plane of the two lines happens 
also to be the horizontal projecting plane. Thecase is not inde- 
terminate, however, as the vertical projection locates the point 
M in space. The reverse of this is also true, that is, the vertical 




Fig. 86. 



Fig. 87. 



Fig. 88. 



instead of the horizontal projections may be coincident. General 
cases of the above are represented pictorially in Figs. 87 and 88. 
Should the horizontal and the vertical projections be coin- 
cident, the lines do not intersect but are themselves coincident 
in space and thus form only one line. 

703. Projection of lines not intersecting in space.* 

There are two possible cases of lines that do not intersect in space. 




Fig. 90. 



The case in which the lines are parallel to each other has previously 
been discussed (701). If the two lines cannot be made to lie 
in the same plane, they will pass each other without intersecting. 
Hence, if in one plane of projection, the projections intersect, 
they cannot do so in the corresponding projection. 

This fact is depicted in Fig. 89. AB and CD are the two 
* Called skew lines. 



CONSIDERATIONS OF LINES AND PLANES 



91 



lines in space. Two distinct points E and F, on the lines are 
shown in the horizontal projection as e and f; their vertical 
projections are, however, coincident. Similarly, G and H are 
also two distinct points on the lines, shown as g' and h' in the 
vertical projection, and coincident, as g and h in the horizontal 
projection. The pictorial representation is given in Fig. 90. 

704. Projection of lines in oblique planes. When a 
third plane is inclined to the principal planes, it cuts them in 
lines of intersection, known as traces (601). Any line, when 
inclined to the principal planes will pierce them in a point. Hence, 
if a plane is to contain "a given line, the piercing points of the 
line must lie in the traces of the plane. Viewing this in another 
way, a plane may be passed through two parallel or two inter- 
secting lines. On the resulting plane, any number of lines may 
be drawn, intersecting the given pair. Hence, an inchned line 
must pass through the trace, if it is contained in the plane.* 

Fig. 91 shows a plane tTt' indicated by its horizontal trace 
tT and its vertical trace Tt'. 
It is required to draw a line AB 
in this plane. Suppose the hori- 
zontal piercing point is assumed 
at b, its corresponding vertical 
projection will lie in the ground 
line (509). Also, if its vertical 
piercing point is assumed at a', 
its corresponding horizontal pro- 
jection will be a. With two 
horizontal projections of given 
points on a line and two vertical 

projections, the direction of the line is determined, for, if the 
horizontal and vertical projecting planes be erected, their inter- 
section determines the given line (610).t 

As a check on the correctness of the above, another line CD 
may be assumed. If the two lines lie in the same plane and are 
not parallel, they must intersect. This point of intersection 

* Whon tho plaiK^ is parallol to iho ground lino, a lino in this i)lano parallol 
to tho princii)al pianos cannot picroo in tho tracos of tho j)lano. So(^ Art. (iO'J. 

t It must be rememborod that the principal pianos must be at right 
angles to each other to determine* this intersection. In the revolved position, 
the planes would not intersect in Iho required lino. 




Fig. 91. 



92 



GEOMETRICAL PROBLEMS IX PROJECTION 



is M shown horizontally projected at m and vertically, at m'. 
The line joining the two is perpendicular to the ground line (508). 
This is illustrated in oblique projection in Fig. 92. 

It should be here noted that if the line is to be contained by 
the plane, only the direction of one projection can be assumed, 
and that the corresponding projection must be found by the 
principles so far developed. 

A converse of this problem is to draw a plane so that it shall 
contain a given line. As an unlimited number of planes can be 
passed through any given line, the direction of the traces is not 
fixed. Suppose AB (Fig. 91) is the given line, then through the 
horizontal piercing point b draw any trace tT, and from T where 
this line intersects the ground line, draw Tt', through the vertical 




Fig. 92. 



piercing point a'. The point T may be located anjrwhere along 
XY. All of these planes will contain the line AB, if their traces pass 
through the piercing points of the line. 

Still another feature of Fig. 91 is the fact that from it can be 
proved that two intersecting lines determine a plane. If AB 
and CD be the two lines intersecting at M, their horizontal piercing 
points are b and c and their vertical piercing points are a' and d' 
respectively. Two points fix the direction of a line, and, hence, 
the direction of the traces is fixed; tT is drawn through cb and 
Tt' is draTVTL through a'd'. The check hes in the fact that both 
traces intersect at one point T on the ground line (606). 

705. Projection of lines parallel to the principal planes 
and lying in an oblique plane. If a line is horizontal, it is 
parallel to the horizontal plane, and its vertical projection must be 



CONSIDERATIONS OF LINES AND PLANES 



93 



parallel to the ground line (511). If this line hes in an oblique 
plane, it can have only one piercing point and that with the vertical 
plane, since it is parallel to the horizontal. Thus, in Fig. 93, 
let tTt' be the given oblique plane, represented, of course, by its 
traces. The given line is AB and a'b' is its vertical projection, 
the piercing point being at a'. The corresponding horizontal 
projection of a' is a. When a line is horizontal, as in this case, 
it may be considered as being cut from the plane tTt by a hori- 
zontal plane, and, as such, must be parallel to the principal 
horizontal plane. These two parallel horizontal planes are cut 
by the given oblique plane tTt' and, from geometry, their lines 
of intersection are parallel. Hence, the horizontal projection 




Fig. 93. 



of the horizontal line must be parallel to the horizontal trace, 
of the given plane because parallel lines in space have parallel 
projections. Accordingly, from a, draw ab, parallel to Tt, and 
the horizontal line AB is thus shown by its projections. 

A line, parallel to the vertical plane, drawn in an oblique plane 
follows the same analysis, and differs only in an interchange of 
the operation. That is to say, the horizontal projection is then 
parallel to the ground line and pierces in a point on the horizontal 
trace of the given plane; its vertical projection must be parallel 
to the vertical trace of the given plane. In Fig. 03, CD is a line 
parallel to the vertical plane and cd is the horizontal projection, 
piercing the horizontal plane at c vc^rtically projected at c'; c'd' 
is, therefore, the required vertical projection. 



94 



GEOMETRICAL PROBLEMS IN PROJECTION 



A check on the problem Hes in the fact that these two lines 
must intersect because they lie in the same plane by hypothesis. 
The point of intersection M is shown as m in the horizontal 




Fig. 94. 



projection, and as m' in the vertical projection. The line joining 
these points is perpendicular to the ground lineo The obUque 
projection of this is shown in Fig. 94. 

Fig. 95 is a still further step in this problem. Three lines 




Fig. 95. 



AB, CD, and EF are shown, all being in the plane tTt'. AB is 
parallel to the vertical plane, CD is parallel to the horizontal 
plane and EF is any other line. It will be observed that the 
three lines intersect so as to form a triangle MNO shown by 
having the area shaded in both its projections. 



CONSIDERATIONS OF LINES AND PLANES 



95 



706. Projections of lines perpendicular to given planes. 

If a line is perpendicular to a given plane, the projections of the 
line are perpendicular to the corresponding traces of the plane. 

Let, in Fig. 96, LL be any plane and MM any other plane 
intersecting it in the trace tr. Also, let AB be any line, perpen- 
dicular to MM, and ab be the projection of AB on LL. It is 
desired to show that the projection ab is perpendicular to the 
trace tr. Any plane through AB is perpendicular to the plane 
MM, because it contains a line perpendicular to the plane by 
hypothesis. Also, any plane through Bb, a perpendicular to 
the plane LL, is perpendicular to LL. Hence, any plane contain- 




FiG. 96. 

ing both lines (it can do so because they intersect at B) will be 
perpendicular both to LL and MM. As the plane through 
ABb is perpendicular to the two planes LL and MM, it is also 
perpendicular to a line common to the two planes, such as tr. 
Thus, ab is perpendicular to tr. In fact, any line perpendicular 
to a plane will have its projection on any other plane perpendicular 
to the trace of the plane, because, instead of LL being assumed 
as the plane of projection and BA a perpendicular to another plane 
MM, the conditions may be reversed and MM be assimied as 
the plane of projection and Bb the perpendicular. 

The converse of this is also true. If Ac be assumed the pro- 
jection of Bb on MM, a plane passed through tlio Vmv Ac ]wv- 
pendicular to the trace tr will contain the lines AB and Bb, wIumv 



96 GEOMETRICAL PROBLEMS IN PROJECTION 

Bb is the given line and AB then the projecting perpendicular 

to the plane MM. 

707. Revolution of a point about a line. Frequently it 
is desirable to know the relation of a point with respect to a line, 
for. if the line and point are given by their projections, the true 
relation may not be apparent. To do this, revolve the point 
about the Ime so that a plane through the point and the line 
will either coincide or be parallel to the plane of projection, then 
they will be projected in their true relation to each other. The 
actual distance between the point and the line is then shown 
as the perpendicular distance from the point to the line. 

Consider the diagram in Fig. 97* and assume that the point 
A is to be revolved about the point B. The projection of A on 





V 




c 


_^ :^ 






1 7:\ 


rr 





a ■ B a a" 

V 


LL 



Fig. 97. 

the line Oa" is a and. to an observer looking down from above 
the line Oa", the apparent distance between A and B is aB. Oa' 
is the apparent distance of A from B to an observer, looking 
orthographically. from the right at a plane perpendicular to 
the plane of the paper through Oa'. Xeither projection gives 
the true relation between A and B from a single projection. If 
the point A is revolved about B as an axis, with BA as a radius, 
until it coincides with the line Oa". A will either be foimd at a" 
or at b!", depending upon the chrection of rotation. During the 
revolution, A always remains in the plane of the paper and de- 
scribes a circle,, the plane of which is perpendictilar to the axis, 
through B. the centre of the circle. 

* This diagram is a profile plane of the given point and of the principal 
planes. 



CONSIDEEATIONS OF LINES AND PLANES 97 

A more general case is shown in Fig. 98 where BB is an axis 
lying in a plane, and A is any point in space not in the plane 
containing BB. If A be revolved about BB as an axis, it will 
describe a circle, the plane of which will be perpendicular to 
the axis. In other words, A will fall somewhere on a line Ba'' 
perpendicular to BB. The line Ba'' must be perpendicular to 
BB because it is the trace of a perpendicular plane (706). This 
point is di" and Ba'' is equal to the radius BA. Contrast this 




Fig. 98. 

with a, the orthographic projection of A on the plane containing 
BB. Evidently, then, a is at a lesser distance from BB than a''. 
Indeed, BA equals Ba'' and is equal to the hjrpothenuse of a 
right triangle, whose base is the perpendicular distance of the 
projection of the point from the axis, and whose altitude is the 
distance of the point above the plane containing the line. The 
angle AaB is a right angle, because a is the orthographic projection 
of A. 

QUESTIONS ON CHAPTER VII 

1. If two lines in space are parallel, prove that their projections on any 

plane are parallel. 

2. When two lines in space intersect, prove that their projections, on 

any plane, intersect. 

3. Draw two lines in space that are not parallel and still do not intersect. 

4. Show a case of two non-intersecting lines whose horizontal projections 

are parallel to each other and whose vertical projections intersect. 
Show also, that the horizontal project inj^ planes of these lines are 
parallel. 



98 GEOMETRICAL PROBLEMS IN PROJECTION 

5. Make an oblique projection of the lines considered in Question 4. 

6. Prove that when a line hes in a plane it must pass through the traces 

of the plane. 

7. Given one projection of a line in a plane, find the corresponding pro- 

jection. 

8. Given a plane, draw intersecting hues in the plane and show by the 

construction that the point of intersection satisfies the ortho- 
graphic representation of a point. 

9. Show how two intersecting lines determine a plane bj" aid of an 

obhque projection. 

10. In a given obhque plane, draw a hne parallel to the horizontal plane 

and show by the construction that this line pierces the vertical 
plane only. Give reasons for the construction. 

11. In Question 9, draw another hne parallel to the vertical plane and 

show that this second line intersects the first in a point. 

12. Given an obhque plane, draw three lines; one parallel to the hori- 

zontal plane, one parallel to the vertical plane and the last inclined 
to both planes. Show, b}^ the construction, that the three lines 
form a triangle (or meet in a point in an exceptional case). 

13. Prove that when a hne is perpendicular to a plane the projection of 

this hne on srny other plane is perpendicular to the trace of the 
plane. Show the general case and also an example in orthographic 
projection. 

14. A hne hes in a given plane and a point is situated outside of the 

plane. Show how the point is revolved about the hne until it is 
contained in the plane. 

15. Prove that the point, while revolving about the hne, describes a 

circle the plane of which is perpendicular to the axis (the line about 
which it revolves). 



CHAPTER VIII 
PROBLEMS INVOLVING THE POINT, THE LINE, AND THE PLANE 

801. Introductory. A thorough knowledge of the preceding 
three chapters is necessary in order to apply the principles, there 
developed, in the solution of certain problems. The commercial 
application of these problems frequently calls for extended knowl- 
edge in special fields of engineering, and for this reason, the 
application, in general, has been avoided. 

Countless problems of a commercial nature may be used as 
illustrations. All of these indicate, in various ways, the im- 
portance of the subject. In general, the commercial problem 
may always be reduced to one containing the mathematical 
essentials (reduced to points, fines and planes). The solution, 
then, may be accomplished by the methods to be shown sub- 
sequently. 

802. Solution of problems. In the solution of the fofiowing 
problems, three distinct steps may be noted: the statement, the 
analysis, and the construction. 

The statement of the problem gives a clear account of what 
is to be done and includes the necessary data. 

The analysis entails a review of the principles involved, and 
proceeds, logically, from the given data to the required conclusion. 
On completion of the analysis, the problem is solved to all intents 
and purposes. 

The construction is the graphical presentation of the analysis. 
It is by means of a drawing and its description that the given 
data is associated with its solution. It may be emphasized 
again that the drawings are made orthographically, and that the 
actual points, lines and planes are to be imagined. 

By a slight change in the assumed data, the resultant con- 
struction may appear widely (fifferent from the diagrains in the 
book. Here, them, is an opportunity to make s(n'(M*nl con- 
structions for the several assumptions, and to prove tliat all 

99 



100 GEOMETRICAL PROBLEMS IX PROJECTION 

follow the general analysis. The simpler constructions might 
be taken and transformed from orthographic to oblique projec- 
tion; this vaW show the projections as well as the actual points, 
lines and planes in space. By performing this transformation 
(from orthographic to oblique projection), the student will soon 
be able to picture the entire problem in space, ^\ithout recourse 
to any diagrams. 

To bring forcibly to the student's attention the difference 
between the analysis and the construction, it may be well to 
note that the analysis gives the reasoning in its most general 
terms, while the construction is specific, in so far as it takes the 
assumed data and gives the solution for that particular caseonly. 

803. Problem 1. To draw a line through a given point, parallel 
to a given line. 

Analysis. If two lines in space are parallel, their projecting 

planes are parallel and their intersectionswith the principal planes 

are parallel (701). Hence, through the projections of the given 

point, draw lines parallel to the projections of the given fines. 

Construction. Let AB, Fig. 99, be the given line in space, 

represented by its horizontal pro- 
jection ab and its vertical projection 
a'b'. Further, let G be the given 
point, similarly represented by its 
horizontal projection g and its ver- 
tical projection g'. Through the 
horizontal projection g, draw cd 
parallel to ab and through g', draw 
Yjq^ 99. c'd' parallel to a'b'. 

As the length of the line is not 
specified, any line that satisfies the condition of paraUefism is 
permissible. Therefore, CD is the line in space that is parallel 
to AB through the point G. A pictorial representation of this 
is shoT\TL in Fig. 8-i. 

804. Problem 2. To draw a line intersecting a given line 
at a given point. 

Analysis. If two lines in space intersect, they intersect in 
a point that is common to the two lines. Therefore, their pro- 
jecting planes vdW intersect in a line which is the projecting line 
of the given point (702). Hence, through the projections of the 




THE POINT, THE LINE, AND THE PLANE 



101 



given point, draw any lines, intersecting the projections of the 
given lines. 

Construction. Let AB, Fig. 100, be the given line, shown 
horizontally projected as ab and ver- 
tically projected as a'b'. Let, also, 
G be the given point, situated on the 
line AB. As no direction is specified 
for the intersecting line, draw any 
line cd through the horizontal projec- 
tion g and this will be the horizontal 
projection of the required line. 
Similarly, any other line c'd' through 
g' will be the vertical projection of 
the required line. Hence, CD and 

AB are two lines in space, intersecting at the point G. The 
pictorial representation of this case is depicted in Figs. 87 and 88. 




Fig. 100. 



805. Problem 3. To find where a given line pierces the 
principal planes. 

Analysis. If a line is oblique to the principal planes, it will 
pierce each of these in a point, the corresponding projection of 
which will be in the ground line. Hence, a piercing point in any 
principal plane must be on the projection of the line in that 
plane. It must also be on a perpendicular erected at the point 
where the corresponding projection crosses the ground line. 
Therefore, the required piercing point is at their intersection. 
Construction. Let AB, Fig. 101, be a limited portion of an 
indefinite line, shown by its horizontal 
projection ab and its vertical projection 
a'b'. The portion chosen AB will not 
picM'ce the principal planes, but its continua- 
tion, in both directions, will. Prolong the 
vertical projection a'b' to c' and at c', erect 
a perpendicular to XY, the ground lin(\ and 
continue it, until it intersects the ])rolonga- 
tion of ab at c. This will be the horizontal 
pi(M-('ing ])()int. In the same way, prolong 
ab to d, ai d, erect a per])endicular to tlie 
tli(^ intersection of which witli the prolonga- 
give the vertical piercing point. II(^ni'i\ 




Fhj. 101, 



ground line as dd 
tion of a'b', at d' wi 



102 



GEOMETRICAL PROBLEMS IN PROJECTION 



if CD be considered as the line, it \sill pierce the horizontal plane 
of projection at c and the vertical plane of projection at d'. A pic- 
torial representation in obHque projection is shown in Fig. 52. 

806. Problem 4. To pass an oblique plane, through a given 
oblique line. 

Analysis. If a plane is obhque to the principal planes, it 
must intersect the ground line at a point (606); and if it is to 
contain a line, the piercing points of the line must lie in the traces 
of the plane (704). Therefore, to draw an oblique plane containing 
a given oblique line, join the piercing points of the line vrith. any 
point of the ground line and the resulting lines will be the traces 
of the required plane. 

Construction. Let AB, Fig. 102, be the given line. This 





Fig. 102. 



Fig. 103. 



line pierces the horizontal plane at a and the vertical plane at b'. 
Assume any point T on the ground line XY, and join T with a 
and also ^ith b'. Ta and Tb' are the required traces, indicated, 
as usual, by tTt'. Thus, the plane T contains the line AB. The 
construction in oblique projection is given in Fig. 103. 

807. Special cases of the preceding problem. If the 

line is parallel to both planes of projection, the traces of the plane 
will be parallel to the ground line (602), and the profile plane 
may be advantageously used in the drawing. 

If the line is parallel to onl}- one of the principal planes, join 
the one piercing point Tvith any point on the ground line, which 
results in one trace of the required plane. Through the point 
on the ground line, draw the corresponding trace, parallel to the 



THE POINT, THE LINE, AND THE PLANE 



103 



projection of the line in the plane containing this trace, 
case is evidently that considered in Art. 705. 



The 



Problem 5. To pass an oblique plane, through a given 
point. 

Analysis. If the oblique plane is to contain a given point, it 
will also contain a line through the point. Hence, through the 
given point, draw an oblique line and find the piercing points 
of this line on the principal planes. Join these piercing 
points with the ground line and the result will be the required 
traces of the plane. 

Construction. Let C, Fig. 104, be the required point, and 
let AB be a line drawn through this point. AB pierces the hori- 





FiG. 104. 



Fig. 105. 



zontal plane at a and the vertical plane at b'. Join a and b' 
with any assumed point T and tTt' will be the required traces. 
The plane T, contains the point C, because it contains a line AB 
through the point C. Fig. 105 shows this pictorially. 

Note. An infinite number of planes may be passed through 
the point, hence the point T was assumed. It might have been 
assumed on the opposite side of the point and would still have 
contained the given point, or the auxiliary line through the 
point. 

Also, it is possil)le to draw a Hue through the gi\en point 
parallel to the ground line. A profile construction will then be 
of service (602). 

Again, a perpendicular line may be drawn through the given 
point and a plane he passed so that it is perpendicular to one or 
both planes of projection (603, 604). 



104 



GEOMETRICAL PROBLEMS IN PROJECTION 



809. Problem 6. To find the intersection of two planes, 
oblique to each other and to the principal planes. 

Analysis. If two planes are oblique to each other, they 
intersect in a line. Any line in a plane must pass through the 
traces of the plane. As the line of intersection is common to the 
two planes, it must pass through the traces of both planes and 
hence it passes through the intersection of these traces. 

Construction. Let T and S, Fig. 106, be the given planes. 
The horizontal piercing point of their line of intersection is at b, 





Fig. 106. 



Fig. 107. 



vertically projected at b'; the vertical piercing point is at a', 
horizontally projected at a. Join ab and a'b', as they are the 
projections of the line AB, which is the intersection of the planes 
T and S. In oblique projection, this appears as shown in Fig. 107. 

810. Special case of the preced= 
ing problem. If the two planes are 
chosen so that the traces in one plane 
do not intersect within the limits of 
the drawing, then draw an auxiliary 
plane R (Fig! 108) and find the inter- 
section, AB as shown. From c' draw 
c'd', parallel to a'b' and from c, draw 
cd, parallel to ab. The line of inter- 
section of the given planes is thus deter- 
mined. 




Fig. 108. 



811. Problem 7. To find the corresponding projection of a 
given point lying in a given oblique plane, when one of its pro- 
jections is given. 

Analysis. If a line lies in a given plane and also contains 



THE POINT, THE LINE, AND THE PLANE 



105 



a given point, the projections of this line will also contain the 
projections of the given point. Hence, through the projection 
of the given point, draw the projection of a line lying in the given 
plane. Then find the corresponding projection of the line. 
The required projection of the given point will lie on the inter- 
section of a perpendicular to the ground line, through the given 
projection of the point with the corresponding projection of the 
line. 

Construction. Let tTt', Fig. 109, be the traces of the given 
plane and c the horizontal projection of the given point. Draw 
ab, the horizontal projection of a line in the plane, through c 
the horizontal projection of the given point. The horizontal 



/ A' 





Fig. 109. 



Fig. 110. 



piercing point of the line AB is in the trace Tt at a, and its cor- 
responding projection lies in the ground line at a'. Further, 
the vertical piercing point lies on a perpendicular to the ground 
line from the point b and also in the trace Tt', hence, it is at b' 
and a'b' is thus the corresponding projection of the line ab. The 
required projection of the given point lies on a line through 
c, perpendicular to the ground line, and also on a'b'; hence, 
it is at their intersection c'. The point C, in space, is con- 
tained in the plane T, and c and c', are corresponding pro- 
jections. The oblique projection of this problem is giv(Mi in 
Fig. 110. 

812. Special case of the preceding problem. The point 
in th(^ above problem was purposely chosen in the first angli\ in 
ordcT to obtain a simi)le case. It may be located anv\vher(\ 



106 GEOMETRICAL PROBLEMS IN PROJECTION 

however, because the planes are indefinite in extent. For instance. 

in Fig. Ill the vertical projection is selected below the ground 

line. However, a single projection does not locate a point in 

space. It may be assumed as Mng either in the third or fourth 

angles (515). Subsequent operations are dependent upon the 

^ ^^ angle in which the point is chosen. 

''\ / Assume, for instance, that the point 

V y^^ is in the fourth angle; the traces of 

\ ^ y . the given plane must then also be 

^ ] ""■^^^''^x ^-^-^^ assumed as being in the fourth angle. 

I ,.^>;><^\ I Thus, Tvdth the given plane T, Tt is 

)^^ ^ "^^6 the horizontal trace, and Tt'" is the 

1^ i vertical trace (609). In completing 

Pjq 111^ the construction by the usual method, 

let c' be the assumed vertical pro- 
jection, and through it, draw a'b' as the vertical projection of 
the assumed line through the gi\'^n point and lying in the given 
plane. This line pierces the horizontal plane at b and the vertical 
plane at a'. Hence, ab and a'b' are the corresponding projections 
of the line AB in space which is situated in the fourth angle. 
The corresponding horizontal projection of c' is c, and thus the 
point C in space is determined. 

Had the point been assumed in the third angle, then the 
traces Tt" and Tt"' w^ould have been the ones to use, Tt" being 
the horizontal, and Tt'" the vertical trace. The construction 
would then, in general, be the same as the previous. 

813. Problem 8. To draw a plane which contains a given 
point and is parallel to a given plane. 

Analysis. The traces of the required plane must be parallel 
to the traces of the given plane. A line may be drawn through 
the given point parallel to an assumed line in the given plane. 
This line wdll then pierce the principal planes in the traces of the 
required plane. Hence, in the given plane, draw any line. 
Through the given point, draw a line parallel to it, and find the 
piercing points of this line on the principal planes. Through 
these piercing points draw the traces of the required plane parallel 
to the corresponding traces of the given plane. The plane so 
draT\Ti is parallel to the given plane. 

Construction. Let T, Fig. 112 be the given plane, and G 



THE POINT, THE LINE, AND THE PLANE 



107 



the given point. In tiie plane T, draw any line CD as shown by 
cd and c'd' its horizontal and vertical projections respectively. 
Through G, draw AB, parallel to CD and ab and a'h' tv411 be 
the projections of this line. The piercing points are a and b' 
on the horizontal and vertical planes respectively. Draw b'S 
parallel to t'T and aS parallel to tT, then sSs' will be the traces 




Fig. 113. 



of the required plane, parallel to the plane T and containing a 
given point G. A check on the accuracy of the construction is 
furnished by having the two traces meet at S. Also, only one 
plane will satisfy these conditions because the point S cannot 
be selected at random. This construction is represented picto- 
rially in Fig. 113. 



\t' 



814. Problem 9. To draw a line perpendicular to a 
plane through a given point. 

Analysis. If a line is perpendicular 
to a given plane the projections of the 
line are perpendicular to the corresponding 
traces of the plane. Hence, draw through 
each projection of the given point, a 
line perpendicular to the corresponding tra('(^ 
(706). 

Construction. Let T, Fig. 114, be 
the given plane, and C, the given point. 
Through c, the horizontal projection of 
draw ab, perpendicular to Tt; and through c 
jection of the given point, draw a'b', 



given 



t— t- 



^ 



Fi< 



111. 



the given pomt, 
the vertical pro- 
perpendicular to Tt'. 



108 



GEOMETRICAL PROBLEMS IN PROJECTION 



Thus, AB is perpendicular to the plane T. An oblique projection 
of this problem is given in Fig. 115. 

815. Special case of the preceding problem. If the 

point is chosen in the third angle, then it must be observed that 
Tt" is the horizontal trace (Fig. 116) and Tt"' is the vertical 
trace. Again, the line AB is drawn perpendicular to T, by- 
making a'b', through c', perpendicular to Tt''', and ab, through c, 
perpendicular to Tt". 

816. Problem 10. To draw a plane through a given point 
perpendicular to a given line. 

Analysis. The traces of the required plane must be perpen- 







\ 


V 
6' 


^ 


\ 






>^ 


a t" 


X 


\ 

V 


\ 


/'^^I'M 1 


'%. 




/' ^*6 




■w. 













Fig. 115. 



Fig. 116. 



dicular to the corresponding projections of the given line (706). 
Through one projection of the given point, draw an auxiliary line 
parallel to the trace; the corresponding projection of this line 
vdW be parallel to the ground line, because it is a line parallel to 
that plane in which its projection is parallel to the trace 
(705). Find the piercing point of this auxihary line, and 
through this point, draw a line perpendicular to the corre- 
sponding projection of the given line. Where this intersects 
the ground Une, draw another line, perpendicular to the 
corresponding projection of the given line. The traces are thus 
determined. 

Construction. Let, in Fig. 117, AB be the given line, and 
C the given point. For convenience, assmne a horizontal line 
in the plane as the auxihary line. Then, through c, draw cd, 



« 



THE POINT, THE LIXE, AND THE PLANE 



109 



perpendicular to ab, and through c', draw c'd', parallel to the 
ground line. The piercing point of this auxiliary line is d', and 
only one point on either trace is required. Hence, through d', 
draw Tt', perpendicular to a'b', and from T, draw Tt, per- 
pendicular to ab. T is, therefore, the required plane, and Tt 
must be parallel to cd because both must be perpendicular to 
ab. Fig. 118 shows a pictorial representation of the same 
problem. 

Note. Instead of having assumed a line parallel to the 
horizontal plane, a line parallel to the vertical plane might have 
been assumed. In the latter case, the vertical projection would 
have been perpendicular to the vertical projection of the line, 




Fig. 11' 



Fig. 118. 



and the horizontal projection would therefore have been parallel 
to the ground line. Also, a point in the horizontal plane would 
have fixed the traces, instead of a point in the vertical plane as 
shown in the problem. 



817. Problem 11. To pass a plane through three given 
points not in the same straight line. 

Analysis. If two of the points be joined by a line, a plane 
may be passed through this line and revolved so that it contains 
the third point. In this position, the plane will contain a Vuw 
joining the third point with any ])oint on the first fine. Hence, 
join two points by a line, and from any point on this line, draw 
another line, through the remaining point. Find the piercing 
points of theso two fines, and thus estal)lish the traces of llie 
reciuircd plane. 



iro 



GEOMETRICAL PROBLEMS IN PROJECTION 



Construction. Let AB and C, in Fig. 119, be the three given 
points. Join A and B and find where this fine pierces the prin- 
cipal planes at d and e'. Assume any point H, on the line AB, 
and join H and C; this line pierces the principal planes at g and 
f. Join dg and f'e' and the traces obtained are those of the 
required plane. A check on the accuracy of the work is furnished 
by the fact that both traces must meet at one point on the ground 
line, as shown at T. Hence, the plane T contains the points 
A, B and C. Fig. 120 is an oblique projection of this problem. 

Note. In the construction of this and other problems, it 
may be desirable to work the problem backwards in order to 




Fig. 119. 



Fig. 120. 



obtain a simpler drawing. It is quite difficult to select three 
points of a plane, at random, so that the traces of the plane 
shall meet the ground line T\dthin the limits of the drawing. In 
working the problem backwards, the traces are first assumed, 
then, any two distinct lines are drawn in the plane, and, finally, 
three points are selected on the two assumed fines of the plane. 
It is good practice, however, to assume three random points 
and proceed with the problem in the regular way. Under these 
conditions, the piercing points are liable to be in any angle, 
and, as such, furnish practice in angles other than the first. 

818. Problem 12. To revolve a given point, not in the 
principal planes, about a line lying in one of the principal planes. 

Analysis. If a point revolves about a line, it describes a 
circle, the plane of which is perpendicular to the axis of revolu- 



THE POINT, THE LINE, AND THE PLANE 111 

tion. As the given line is the axis, the point will fall somewhere 
in the trace of a plane, through the point, perpendicular to the 
axis. The radius of the circle is the perpendicular distance from 
the point to the line; and is equal to the hypothenuse of a tri- 
angle, whose base is the distance from the projection of the point 
to line in the plane, and whose altitude is the distance of the 
corresponding projection of the point from the plane containing 
the line (707). 

Construction. Assume that the given line AB, Fig. 121, 
lies in the horizontal plane and therefore is its own projection, 
ab, in that plane; its corresponding projection is a'b' and lies 
in the ground line. Also, let C be the ^ 

given point, sho^n by its projections c ^\ 

and c'. Through c, draw cp, perpendic- 1 \ 

ular to the line ab; cp is then the trace 
of the plane of the revolving point and 
the revolved position of the point will fall 
somewhere along this line. The radius of 
the circle is found by making an auxiliary 
view, in which c'o is the distance of the 
point above the horizontal plane, and 

oq = cp is the distance of the horizontal projection of the given 
point from the axis. Hence, c'q is the radius of the circle, and, 
therefore, lay off pc" = c'q. The revolved position of the point 
C in space is, therefore, c". The distance pc'' might have been 
revolved in the opposite direction and thus would have fallen 
on the opposite side of the axis. This is immaterial, however, 
as the point is always revolved so as to make a clear diagram. 

Note. In the construction of this problem, the line was 
assumed as lying in the horizontal plane. It might have been 
assumed as lying in the vertical plane, and, in this case, the 
operations would have been identical, the only difference being 
that the trace of the plane containing the path of the point would 
then lie in the vertical plane. The auxiliary diagram for deter- 
mining the radius of the circle may be constructed ])elow the 
ground line, or, if desirable, in an entirely separate diagram. 

819. Problem 13. To find the true distance between two 

points in space as given by their projections. First method. 

Analysis. Tlic true distance is e(]ual to the K'ngtli of the 




112 



GEOMETRICAL PROBLEMS IN PROJECTION 



line joining the two points. If, then, one projecting plane of 
the line be revolved until it is parallel to the corresponding plane 
of projection, the line will be shown in its true length on the 
plane to which it is parallel. 

Construction. Case 1. When both points are above the 
plane of projection. — Let AB, in Fig. 122, be the given line. 
For convenience, revolve the horizontal projecting plane about 
the projecting perpendicular from the point A on the line. The 
point B will describe a circle, the plane of which is perpendicular 
to the axis about which it revolves. As the plane of the circle 
(or arc) is parallel to the horizontal plane, it is projected as 
the arc be. The corresponding projection is b'c', because its 





Fig. 122. 



Fig. 123. 



plane is perpendicular to the vertical plane. In the position 
ac, the projecting plane of AB is parallel to the vertical plane, 
and c' is the vertical projection of the point b, when so 
revolved. Hence, a'c' is the true length of the line AB in 



space, 



820. Case 2. When the points are situated on opposite sides 
of the principal plane. — Let ABy in Fig. 123, be the given line. 
Revolve the horizontal projecting plane of the line about the 
horizontal projecting perpendicular from the point A on the 
line. The point B will describe an arc which is horizontally pro- 
jected as be and vertically projected as b'e'. The ultimate 
position of b' after revolution is at e', whereas a' remains fixed. 
Hence, a'e' is the true length of the line AB. 



THE POIXT, THE LINE, AND THE PLANE 



113 



821. Problem 13. To find the true distance between two 
points in space as given by their projections. Second method. 

Analysis. The true distance is equal to the length of a 
hne joining the two points. If a plane be passed through the 
line and revolved about the trace into one of the principal planes, 
the distance between the points will remain unchanged, and in 
its revolved position, the line will be shown in its true length. 

Construction. Case 1. When both points are above the 
plane of projection. — Let A and B Fig. 124 be the two points in 
question. For convenience, use the hori- ^, 
zontal projecting plane of the line as the 
revolving plane ; its trace on the horizontal 
plane is ab, which is also the projection 
of the line. The points A and B, while 
revolving about the line ab, will describe 
circles, the planes of which are perpendic- 
ular to the axis, and, hence, in their re- 
volved position, will lie along lines aa" 
and bb''. In this case, the distance of the 
projections of the points from the axis is 
zero, because the projecting plane of the 
line joining the points was used. The 

altitudes of the triangles are the distances of the points above 
the horizontal plane, and, hence, they are also the hypothenuses 
of the triangles which are the radii of the circles.* Therefore, 
lay off aa'' = a'o and bb'' = b'p along lines aa" and bb'', which 
are perpendicular to ab. Hence, a''b'' is the true distance 
between the points A and B in space. 

822. Case 2. When the points are situated on opposite 
sides of the principal plane. — Let AB, 
Fig. 125, be the two points. From the 
projections it will be seen that A is in 
the first angle and B is in the fourth 
angle. If, as in the previous case, the 
horizontal projecting plane is used as 
the revolving plane then the horizontal 
projection ab is the trace of the revolv- 
ing ])lane as before. The point A falls 

Similnrlv, B falls to b" wheiv bb" = b'p, 





Fic;. 125. 



to a' 



where aa'' = a'o. 

* Sec the soinowliat .similar case in ProbliMu 1 



114 



GEOMETKICAL PROBLEMS IN PROJECTION 



but, as must be noticed, this point falls on that side of ab, oppo- 
site to the point a''. A little reflection will show that such must 
be the case, but it may be brought out by what follows: The 
line AB pierces the horizontal plane at m and this point must 
remain fixed during the revolution. That it does, is shown by 
the fact that the revolved position of the hne a"b" passes 
through this point. 

Note. In both cases, the vertical projecting plane might 
have been used as the revolving plane and after the operation 
is performed, the true length of the line is again obtained. It 
must of necessity be equal to that given by the method here 
indicated. 

823. Problem 14. To find where a given line pierces a 
given plane. 

Analysis. If an auxiliary plane be passed through the 




Fig. 126. 



Fig. 127. 



given line, so that it intersects the given plane, it will cut from 
it a line that contains the given point. The given point must 
also lie on the given line, hence it lies on their intersection. 

Construction. Let T, Fig. 126, be the given plane and AB 
the given line. For convenience, use the horizontal projecting 
plane of the line as the auxiliary plane; the horizontal trace is 
cb and the vertical trace is cc'. The auxiliary plane cuts from 
the plane T the line CD. The vertical projection is shown as 
c'd' and the horizontal projection cd is contained in the trace 
of the horizontal projecting plane, because that plane was pur- 



THE POINT, THE LINE, AND THE PLANE 115 

posely taken as the cutting plane through the Hne. It is only in 
the vertical projection that the intersection m' is determined; 
its horizontal projection m is indeterminate in that plane (except 
from the fact that it is a corresponding projection) since the 
given line and the line of intersection have the same horizontal 
projecting plane. An oblique projection of this problem is given 
in Fig. 127. 

824. Problem 15. To find the distance of a given point 
from a given plane. 

Analysis. The perpendicular distance from the given point 
to the given plane is the required distance. Hence, draw a per- 
pendicular from the given point to the given plane, and find 
where this perpendicular pierces the given plane. If the line 
joining the given point and the piercing point be revolved into 
one of the planes of projection, the line will be shown in its true 
length. 

Construction. Let T, Fig. 128, be the given plane, and A, 
the given point. From A, draw AB, 
perpendicular to the plane T; the pro- \ i 

jections of AB are therefore perpendicu- 4^' 

lar to the traces of the plane. If the iV X 

horizontal projecting plane of the line I \\( 

AB be used as the auxiliary plane, it ^ ^1 Ty\d\.T 

cuts the given plane in the line CD, axI^ 

and pierces it at the point B. Revolve v ^^\ 

the projecting plane of AB about its /A ^ 

horizontal trace ab into the horizontal ^ \ 7°" 

plane of projection. A will fall to a", \ / 

where aa" = a'o, and B will fall to b", \/ 

where bb'' = b'p. Therefore, a''b'' is the 
distance from the point A to the plane T. ^^^' 128. 

825. Problem 16. To find the distance from a given point 
to a given line. 

Analysis. Through the given point pass a plane perpen- 
dicular to the given line. The distance between the piercing 
point of the given line on this plane and the given point is the 
required distance. Join these two points by a line and revolve 
this line into one of the planes of ])roje('tion; the lin(^ will then 
be seen in its true length. 



116 



GEOMETRICAL PROBLEMS IN PROJECTION 



Construction. Let AB, Fig. 129, be the given line, and G, 
the given point. Through G, draw a plane perpendicular to AB 
(816), by drawing gc perpendicular to ab, and g'c', parallel to 

the ground line; the piercing point 
of this line on the vertical plane 
is c'. Hence, the traces Tt' and 
Tt, perpendicular to a'b' and ab, 
respectively, through the point c', 
will be the traces of the required 
plane. AB pierces this plane at 
B, found by using the horizontal 
projecting plane of AB as a cutting 
plane, this cutting plane inter- 
secting in a line DE. The point 
B must be on both AB and DE. 
The projected distance, then, is 
the distance between the points B 
and G ; the true distance is found 
by revolving BG into the horizontal plane. The latter operation 
is accomplished by using the horizontal projecting plane of BG 
and revolving it about its trace bg; G falls to g'', where gg'' = 
g'p, and B falls to b'^ where bb'' = b'o. Thus, g^'b'' is the true 
distance between the point G and the line AB. 




Fig. 129. 



826. Problem 17. To find the angle between two given 
intersecting lines. 

Analysis. If a plane be passed through these lines and 
revolved into one of the planes of projection, the angle will be 
shown in its true size. Hence, find the piercing points of the 
given lines on one of the planes of projection; the line joining 
these piercing points will be the trace of the plane containing 
the lines. Revolve into that plane and the revolved position 
of the two lines shows the true angle. 

Construction. LetAB and AC, Fig. 130, be the two given 
lines intersecting at A. These lines pierce the horizontal plane 
of projection at b and c and be is the trace of a plane containing 
the two lines. If A be considered as a point revolving about 
the line be, it then describes a circle, the plane of which is per- 
pendicular to be and the point A will coincide with the horizontal 
plane somewhere along the line oa''. The radius of the circle 



THE POINT, THE LINE, AND THE PLANE 



117 



described by A is equal to the hypothenuse of a right triangle, 
where the distance ao, the projec- 
tion of a from the axis, is the base, 
and a'p, the distance of the point 
above the plane, is the altitude. 
This is shown in the triangle a'pq, 
where pq is equal to ao and therefore 
a'q is the required radius. Hence, 
make oa'' = a'q, and a'' is the re- 
volved position of the point A in 
space. The piercing points b and c 
of the given lines do not change 
their relative positions. Thus a''b 
and a''c are the revolved position 
of the given Unes, and the angle ba^'c is the true angle. 




827. Problem 18. To find the angle between two given 
planes. 

Analysis. If a plane be passed perpendicular to the line of 
intersection of the two given planes, it will cut a line from each 
plane, the included angle of which will be the true angle. Re- 
volve this plane, containing the lines, about its trace on the 
principal plane, until it coincides with that plane, and the angle 
will be shown in its true size. 

Construction. Let T and S, Fig. 131, be the two given 
planes, intersecting, as shown, in the line AB. Construct a sup- 
plementary plane to the right of the main diagram. H'H' is 
the new horizontal plane, shown as a line parallel to ab. The 
line AB pierces the vertical plane at a distance a'a above the 
horizontal plane. Accordingly, a'a is laid off on V'V, perpendicu- 
lar to H'H'; it also pierces the horizontal plane at b sho\Mi in both 
views. The supplementary view shows ba' in its true relation 
to the horizontal plane, and is nothing more or less than a side 
view of the horizontal projecting plane of the line AB. If, in 
this supplementary view, a ]XM])endicular plane cd be drawn, 
it will intersect the line AB in c, and the horizontal plane in the 
trace dfe shown on (^d. The lettering in both views is such 
that similar letters indicate similar points. IIenc'(\, dfe is the 
trace of the plane as shown in \\w main diagram, and ec and 
dc are the two Vnws cut from the ])lanes T and S l>y tlu^ plan*' 



118 



GEOMETRICAL PROBLEMS IN PROJECTION 



cde. When the plane cde is revolved into coincidence with 
the horizontal plane, c falls to c" in the supplementary view 
and is projected back to the main diagram as c". Therefore, 
ec^'d is the true angle between the planes, because e and d remain 
fixed in the revolution. 

828. Problem 19. To find the angle between a given plane 
and one of the principal planes. 

Analysis. If an auxiliary plane be passed through the given 
plane and the principal plane so that the auxiliary plane is per- 



^yt 




Fig. 131. 



pendicular to the intersection of the given plane and the prin- 
cipal plane, it will cut from each a fine, the included angle of 
which will be the true angle. If, then, this auxiUary plane be 
revolved into the principal plane, the angle will be shown in its 
true size. 

Construction. Let T, Fig. 132, be the given plane. The 
angle that this plane makes with the horizontal plane is to be 
determined. Draw the auxiUary plane R, so that its horizontal 
trace is perpendicular to the horizontal trace of the given plane; 
the vertical trace of the auxiliary plane must as a consequence 
be perpendicular to the ground line as Rr'. A triangle rRr' is 



THE POINT, THE LINE, AND THE PLANE 



119 



cut by the auxiliary plane from the given plane and the two 
principal planes. If this triangle be revolved into the horizontal 
plane, about rR as an axis, the point x' will fall to x" with Rr' 
as a radius. Also, the angle rRr'' must be a right angle, because 
it is cut from the principal planes, which are at right angles to 
each other. Hence, Rrr'' is the angle which the plane T makes 
with the horizontal plane of projection. 

The construction for obtaining the angle with the vertical 
plane is identical, and is shown on the right-hand side, with 
plane S as the given plane. All construction lines are added 
and no comment should be necessary. 

Note. The similarity of Probs. 18 and 19 should be noted. 
In Prob. 19 the horizontal trace is the intersection of the given 




Fig. 132. 



plane and the horizontal plane; hence, the auxiliary plane is 
passed perpendicular to the trace. A similar reasoning applies to 
the vertical trace. 



829. Problem 20. To draw a plane parallel to a given plane, 
at a given distance from it. 

Analysis. The required distance between the two planes 
is the perpendicular distance, and the resulting traces must be 
parallel to the traces of the given plane. If a plane be passed 
perpendicular to either trace, it will cut from the principal planes 
and the given plane a right angled triangle, the hypothonuse 
of which will be the line cut from the given piano. If, further, 
the triangle be revolved into the plane containing the trace and 
the required distance between the planes be laid off perpendicular 
to the hypothonuse cut from the given ])lano, it will establish 
a point on the hypothonuse of the required i)lane. A line parallel 



120 



GEOMETRICAL PROBLEMS IN PROJECTION 



to the hypothenuse through the estabhshed point will give the 
revolved position of a triangle cut from the required plane. On 
the counter revolution, this triangle t\^11 determine a point in 
each plane, through which the required traces must pass. Hence, 
if hnes be drawn through the points so found, parallel to the 
traces of the given plane, the traces of the required plane are 
established. 

Construction. Let T, Fig. 133, be the given plane, and r"g 
the required distance between the parallel planes. Pass a plane 

rOr', perpendicular to the hori- 
zontal trace Tt; its vertical trace 
Or is, therefore, perpendicular 
to the ground line. The re- 
volved position of the triangle 
cut from the two principal planes 
and the given plane is rOr". 
Lay off r'^g, perpendicular to 
xt'\ and equal to the required 
distance between the planes. 
Draw uu" parallel to tt" and 
the triangle cut from the prin- 
cipal planes and the required 
plane is obtained. On counter revolution, u" becomes u' and u 
remains fixed; Su' and Su, parallel respectively to Tr' and Tr, 
are the required traces. Therefore, the distance between the 
planes T and S is equal to T"g. 

830. Problem 21. To project a given line on a given plane. 

Analysis. If perpendiculars be dropped from the given 
line upon the given plane, the points, so found, are the projec- 
tions of the corresponding points on the line. Hence, a line 
joining the projections on the given plane is the required pro- 
jection of the Hne on that plane. 

Construction. Let T, Fig. 134, be the given plane, and 
AB the given line. From A, draw a perpendicular to the plane 
T; its horizontal projection is ac and its vertical projection is 
a'c'. To find where AC pierces the given plane, use the hori- 
zontal projecting plane of AC as the cutting plane; FE is the 
line so cut, and C is the resultant piercing point. Thus, C is 
the projection of A on the plane T. A construction, similar in 




Fig. 133. 



THE POINT, THE LINE, AXD THE PLANE 



121 



detail, will show that D is the projection of B on the plane T. 
Hence, CD is the projection of AB on the plane T. 

831. Problem 22. To find the angle between a given line 
and a given plane. 

Analysis. The angle made by a given line and a given plane 
is the same as the angle made by the given line and its projection 




Fig. 134. 



on that plane. If from any point on the given line, another 
line be drawn parallel to the projection on the given plane, it 
will also be the required angle. The projection of the given line 
on the given plane is perpendicular to a projecting perpendicular 
from the given line to the given plane. Hence, any line, parallel 
to the projection and lying in the projecting plane of the given 
line to the given plane is also perpendicular to this projecting 
perpendicular. Therefore, pass a plane through the given line 
and the projecting perpendicular from the given line to the 
given plane. Revolve this ])lane into one of the ])lan(*s of ])n>- 



122 



GEOMETRICAL PROBLEMS IN PROJECTION 



jection, and, from any point on the line, draw a perpendicular 
to the projecting perpendicular. The angle between this line 
and the given line is the required angle. 

Construction. Let T, Fig. 135, be the given plane, and 
AB the given line. From B, draw a perpendicular to the plane 
T, by making the projections respectively perpendicular to the 
traces of the given plane. Find the piercing points e and f, 
on the horizontal plane, of the given line and this perpendicular. 
Revolve the plane containing the lines BE and BF; B falls to b'', 
on a line b"p, perpendicular to ef. The distance b''p is equal 
to the hj^othenuse of a right triangle, where bp is the base and 



^.< 




Fig. 135. 

b'o is the altitude; b'oq is such a triangle, where oq = bp. Hence, 
b''p is laid off equal to b'q. If from any point d, a line dc be 
drawn, perpendicualr to b'^f, then b^'dc is the required angle, 
as dc is parallel to the projection of AB on the plane T in its 
revolved position. 

832. Problem 23. To find the shortest distance between 
a pair of skew * lines. 

Analysis. The required line is the perpendicular distance 
between the two hues. If through one of the given lines, another 
line be draT\Ti parallel to the other given line, the intersecting lines 
will establish a plane which is parallel to one of the given lines. 



* Skew lines are lines which are not parallel and which do not intersect. 



THE POINT. THE LINE, AND THE PLANE 



123 



The length of a perpendicular from any point on the one given 
line to the plane containing the other given line, is the required 
distance. 

Construction. Let AB and CD, Fig. 136, be the two given 




Fig. 136. 



lines. Through any point O, on AB, draw FE, parallel to CD, 
and determine the piercing points of the lines AB and FE; a, 
e and f , b' are these piercing points, and, as such, tlotcrniino 
the plane T. CD is then parallel to the plane T. From any 
point G, on CD, draw GH, perpendicular to the plane T, and 



124 



GEOMETRICAL PROBLEMS IN PROJECTION 



find its piercing point on that plane. This point is H, found 
by draT\ing gh perpendicular to Tt and g'h' perpendicular to 
Tt'; the horizontal projecting plane cuts from the plane T, a 
line MN, on which is found H, the piercing point. GH is there- 
fore the required distance, but to find its true length, revolve 




Fig. 136. 



the horizontal projecting plane of GH into the horizontal plane. 
On revolution, H falls to h", where hh" is equal to the distance 
h' above the ground line; and, similarly, G falls to g'\ There- 
fore, Wg'' is the true distance between the lines AB and CD. 



THE POINT, THE LINE, AND THE PLANE 



125 



Note. In order to find the point on each of the lines at 
which this perpendicular may be drawn, project the one given 
line on the plane containing the other. Where the projections 
cross, the point will be found. 

Additional Constructions 

833. Application to other problems. The foregoing prob- 
lems may be combined so as to form additional ones. In such 
cases, the analysis is apt to be rather long, and in the remaining 
few problems it has been omitted. The construction of the 




Fig. 137. 



problem might be followed by the student and then an analysis 
worked up for the particular problem afterward. 

834. Problem 24. Through a given point, draw a line of a 
given length, making given angles with the planes of projection. 

Construction. Consider the problem solved, and let AB, 
Fig. 137, be the required line, through the point A. The con- 
struction will first be shown and then the final position of the 
line AB will be analytically considered. From a', draw a'c' 
of a length 1, making the angle a with the ground line, and draw 
ac, parallel to the ground line. The horizontal ]:)rojocting plane 
of AB has been revolved about the horizontal i)roj(H'ting ])orpen- 
dicular, imtil it is parallel to the vertical plane, ami. therefore 



126 



GEOMETRICAL PROBLEMS IX PROJECTION 



a'c' is shown in its true length and inchnation to the horizontal 
plane. On the counter-revolution of the projecting plane of AB, 
it mil be observed that the point A remains fixed, because it 
Hes in the axis; B describes a circle, however, whose plane is 
parallel to the horizontal plane, and is, therefore, projected as 
the arc cb, while its vertical projection (or trace, if it be consid- 
ered as a plane instead of the moving point) is b'c', a line parallel 
to the ground line. The angle that the line AB makes ^dth the 
horizontal plane is now fixed, but the point B is not finally located 
as the remaining condition of making the angle ^ with the hori- 
zontal plane is yet conditional. 

In order to lay off the angle that the line makes with the 




Fig. 137. 



vertical plane, draw a line ad, through a, making the angle ^ 
wdth the ground line, and of a length L From a', draw a'd', 
so that d' is located from its corresponding projection d. Draw 
bd, through d, parallel to the ground line, and where its inter- 
section with the arc cb, locates b, the final position of the hori- 
zontal projection .of the actual line. The corresponding pro- 
jection b', may be located by drawing the arc d'b', and finding 
where it intersects the line b'c', through c', parallel to the ground 
line. The points b and b' should be corresponding projections, 
if the construction has been carried out accurately. In revie^vdng 
the latter process, it is found that the vertical projecting plane 



THE POINT, THE LINE, AND THE PLANE 



127 



of the line has been revolved about the vertical projecting per- 
pendicular through A, until it was parallel to the horizontal 
plane. The horizontal projection is then ad, and this is shown 
in its true length and inclination to the vertical plane. 

It may also be noted that the process of finding the ultimate 
position of the fine is simply to note the projections of the path 
of the moving point, when the projecting planes of the line are 
revolved. That is, when the horizontal projecting plane of the 
line is revolved, the line makes a constant angle with the hori- 
zontal plane; and the path of the moving point is indicated by 
its projections. Similarly, when the vertical projecting plane 
of the fine is revolved, the line makes a constant angle with the 
vertical plane; and the path of the moving point is again given 



U^' }^^ 



Fig. 138. 



by its projections. Where the paths intersect, on the proper 
planes, it is evidently the condition that satisfies the problem. 
There are four possible solutions for any single point in space, 
and they are shown in Fig. 138. Each is of the required length, 
and makes the required angles with the planes of projection. 
The student may try to work out the construction in each case 
and show that it is true. 

835. Problem 25. Through a given point, draw a plane, 
making given angles with the principal planes. 

Construction. Prior to the solution of this problem, it is 
desirable to investigate the property of a line from any point 
on the ground line, perpendicular to the plane. If the angles 
that this line makes with the principal planes can be determined, 
the actual construction of it, in projection, is then similar to the 
preceding problem. To draw the required plane, hence, resolves 
itself simply into passing a plane through a given point, per- 
pendicular to a given line (816). 



128 



GEOMETRICAL PROBLEMS IN PROJECTION 



Let, in Fig. 139, T be the required plane, making the required 
angles a with the horizontal plane, and ^ with the vertical plane. 
To find the angle made with the horizontal plane, pass a plane 
perpendicular to the horizontal trace as AOB, through any as- 
sumed point O on the ground fine; OA is thus perpendicular 
to Tt and OB is perpendicular to the ground line. This plane 
cuts from the plane T, a line AB, and the angle BAO is the required 
angle a. Similarly, pass a plane COD, through O perpen- 
dicular to the vertical trace, then CO is perpendicular to Tt 
and OD is perpendicular to the ground hne; DCO is the required 
angle ^, which the plane T makes with the vertical plane. The 
planes AOB and COD intersect in a Une OP. OP is perpendicular 




Fig. 139. 

to the plane T because each plane. AOB and COD is perpendicular 
to the plane T (since they are each perpendicular to a trace, 
which is a line in the plane), and, hence, the line common to the 
two planes (OP) must be perpendicular to the plane. The 
angles OPC and OPA are, therefore, right angles, and, as a result, 
angle POA = 90°-a, and angle POC = 90°-^. Hence, to 
draw a perpendicular to the required plane, draw a line making 
angles with the principal planes equal to the complements (90° 
minus the angle) of the corresponding angles. It is evident 
that any line will do, as all such lines, when measured in the 
same way, will be parallel, hence, it is not necessary, although 
convenient, that this line should pass through the ground line. 
To complete the problem, let AB, Fig. 140, be a line making 



THE POINT, THE LINE, AND THE PLANE 



129 



angles 90°- a with the horizontal plane, and 90°- g with the 
vertical plane. Through G, the given point, draw a perpendicular 
plane to AB, and the resultant plane T is the required plane 




Fig. 140. 

making an angle a, with the horizontal plane, and an angle p, 
with the vertical plane. 

Note. As there are four solutions to the problem of drawing 
a line making given angles with the principal planes, there are 
also four solutions to this 

problem. The student may t 

show these cases and check the 
accuracy by finding the angle 
between a given plane and the 
principal planes (828). 

836. Problem 26. Through 
a given line, in a given plane, 
draw another line, intersecting 
it at a given point, and at a 
given angle. 

Construction. Let AB, 
Fig. 141, be the given Hue, T 
the given plane and G the 
given point. Revolve the 
limited portion of the plane 
tTt' into coincidence with the 

horizontal plane. Tt remains fixed but Tt' revolves to Tt". 
To find th(^ direction of Tt'', consider any point b' on tlie original 
position of the trace Tt'. The distance Tb' nuist equal Tb" as 




Fig. 14L 



130 



GEOMETRICAL PROBLEMS IX PROJECTION 



this length does not change on revolution; since Tt is the axis of 
revolution, the point B describes a circle, the plane of which is 
perpendicular to the axis, and, therefore, b'' must also He on a 
line bb", from b, perpendicular to the trace Tt. 

In the revolved position of the plane, the point a remains 
unchanged, while B goes to b'^ Hence, ah'' is the revolved 
position of the given line AB. The given point G moves to g", 
on a line gg'^, perpendicular to Tt and must also be on the hne 
ab''. Through g", draw the line cd", making the required angle 
a with it. On counter-revolution, c remains fixed, and d" 
moves to d'. Thus CD is the required line, making an angle 
a with another line AB, and lying in the given plane T. 




Fig. 142. 



837. Problem 27. Through a given line in a giyen plane, 
pass another plane, making a given angle with the given plane. 

Construction. Let T, Fig. 142, be the given plane, AB 
the given line in that plane, and a the required angle between 
the planes. Construct a supplementary view of the line AB; 
H'H' is the new horizontal plane, the inclination of ab' is shown 
by the similar letters on both diagrams. The distance of b' 
above the horizontal plane must also equal the distance b' above 
H'H' and so on. Through h, in the supplementary view, draw 
a plane hf perpendicular to ab'. Revolve h about f to g and 
locate g as shown on the horizontal projection ab, of the main 



THE POINT, THE LINE, AND THE PLANE 



131 



diagram. The line eg is cut from the plane T, by the auxihary 
plane egf, hence, laj^ off the angle a as shown. This gives 
the direction gf of the line cut from the required plane. The 
point f lies on gf and also on ef which is perpendicular to ab. 
Hence, join af and produce to S; join S and b' and thus estab- 




FiG. 143. 



lish the plane S. The plane S passes through the given lino AB 
and makes an angle a with the given plane T (827). 

838. Problem 28. To construct the projections of a circle 
lying in a given oblique plane, of a given diameter, its centre 
in the given plane being known. 

Construction. Let T, Fig. 143, l)e the given i)lane, and G 



132 



GEOMETEICAL PROBLEMS IN PROJECTIOX 



the given point h-ing in the plane T. When g', is assumed, for 
instance, g is found by drawing a horizonta' line g'a' and then 
ga is its corresponding projection; g can therefore be determined 
as shown. When the plane of a circle is inclined to a plane of 




Fig. 143. 



projection, it is projected as an elhpse. An elhpse is determined, 
and can be constructed, when its major and minor axes are given.* 
To construct the horizontal projection, revolve the plane T 
about Tt, until it coincides with the horizontal plane as Tt". 
The direction of Tf is found by drawing aa'' perpendicular to 
Tt and lajdng off Ta' = Ta" (836). The centre of the circle in 

* For methods of constructing the ellipse see Art. 906. 



THE POINT, THE LINE, AND THE PLANE 133 

its revolved position is found by drawing a'^g'' parallel to Tt 
and gg" perpendicular to Tt; g" is this revolved position. With 
the given radius, draw c"&"Q"t'\ the circle of the given diameter. 
Join e"&" and t"c"\ prolong these lines to o and q, and also 
draw g"p parallel to these. Thus, three parallel lines in the 
revolved position of the plane T, are established and on counter 
revohition, they will remain parallel. The direction gp, of one 
of them is known, hence, make fq and eo parallel to gp. The 
points cdef are the corresponding positions of Q,"&"e"i" and 
determine the horizontal projection of the circle. The line ec 
remains equal to q"c" but fd is shorter than i"&'\ hence, the 
major and minor axes of an ellipse (projection of the circle) are 
determined. The ellipse may now be drawn by any convenient 
method and the horizontal projection of the circle in the plane T 
will be complete. 

By revolving the plane T into coincidence with the vertical 
plane, Tt''' is found to be the revolved position of the trace Tt 
and g'", the revolved position of the centre. The construction 
is identical with the construction of the horizontal projection 
and will become apparent, on inspection, as the necessary lines 
are shown indicating the mode of procedure. As a result, kT'm'n' 
determine the major and minor axes of the ellipse, which is the 
vertical projection of the circle in the plane T. 

Note. As a check on the accuracy of the work, tangents 
may be drawn in one projection and the corresponding projection 
must be tangent at the corresponding point of tangency. 



QUESTIONS ON CHAPTER VIII 

1. Mention the three distinct steps into which the solution of a problem 

may be divided. 

2. What is the statement of a problem? 

3. What is the analysis of a problem? 

4. What is the construction of a problem? 

5. What type of projection is generally used in the construction of a 

problem? 

Note. In the following problems, the construction, except in a few 
isolated cases, is to be entirely limited to the first angle of pro- 
jection. 

6. Draw a line through a given point, parallel to a given line, (live 

analysis and construction. 



134 GEOMETRICAL PROBLEMS IX PROJECTION 

7. Draw a line intersecting a given line at a given point. Give analysis 

and construction. 

8. Find where a given oblique line pierces the planes of projection. 

Give analysis and construction. 

9. Transfer the diagram of Question 8 to obhque projection. 

10. Pass an obhque plane through a given obhque line. Give analj^sis 

and construction. 

11. ]\Iake an obhque projection of the chagram in Question 10. 

12. Pass a plane obhquely tlii'ough the principal planes and through a 

line parallel to the ground hne. Give analysis and construction. 
Hint: L'se a profile plane in the construction. 

13. Transfer the diagram of Question 12 to an obhque projection. 

1-4. Pass an obhciue plane through a line which is parallel to the hori- 
zontal plane but inchned to the vertical plane. Give analysis 
and construction. 

15. Transfer the diagram of Question 14 to an obhque projection. 

16. Pass an obhque plane through a hne wliich is parallel to the vertical 

plane but inclined to the horizontal plane. Give analysis and 
construction. 

17. Transfer the diagram of Question 16 to an obhque projection. 

IS. Pass an oblique plane through a given point. Give analysis and 
construction. 

19. ^lake an oblique projection of the diagram in Question 18. 

20. Find the intersection of two planes, obhque to each other and to the 

principal planes. Give analysis and construction. 

21. Transfer the chagram of Question 20 to an obhque projection. 

22. Find the intersection of two planes, oblique to each other and to the 

principal planes. Take the case where the traces do not intersect 
on one of the principal planes. Give analysis and construction. 

23. Find the corresponding projection of a given point Mng in a given 

obhque plane, when one projection is given. Give analysis and 
construction. 

24. Transfer the chagram of Question 23 to an obhcjue projection. 

25. Draw a plane wliich contains a given point and is parallel to a given 

plane. Give analysis and construction. 

26. Transfer the chagram of Question 25 to an obhcj[ue projection. 

27. Draw a hne through a given point, perpenchcular to a given plane. 

Give analysis and construction. 

28. Transfer the chagram of Question 27 to an obhque projection. 

29. Draw a plane through a given point, perpendicular to a given line. 

Give analysis and construction. 

30. Transfer the chagram of Question 29 to an obhque projection. 

31. Pass a plane through three given points, not in the same straight 

line. Give analysis and construction. 

32. Transfer the chagi'am of Question 31 to an oblique projection. 

33. Revolve a given point, not in the principal planes, about a line 

lying in one of the principal planes. Give analysis and construc- 
tion. 



THE POINT, THE LINE, AND THE PLANE 135 

34. Transfer the diagram of Question 33 to an oblique projection and 

show also the plane of the revolving point. 

35. Find the true distance between two points in space, when both 

points are in the first angle of projection. Use the method of 
revolving the projecting plane of the hne until it is parallel to one 
plane of projection. Give analysis and construction. 

36. Make an obhque projection of the diagram in Question 35 and show 

the projecting plane of the hne. 

37. Find the true distance between two points in space, when one point 

is in the first angle and the other is in the fourth angle. Use the 
method of revolving the projecting plane of the line until it is 
parallel to one plane of projection. Give analysis and construction. 

38. Make an oblique projection of the diagram in Question 37 and show 

the projecting plane of the line. 

39. Find the true distance between two points in space, when both 

points are in the first angle of projection. Use the method of 
revolving the projecting plane of the hne into one of the planes 
of projection. Give analysis and construction. 

40. Transfer the diagram of Question 39 to an obhque projection and 

show the projecting plane of the hne and the path described by 
the revolving points. 

41. Find the true distance between two points in space, when one point 

is in the first angle and the other is in the fourth angle of pro- 
jection. Use the method of revolving the projecting plane of 
the line into one of the planes of projection. Give analysis and 
construction. 

42. Transfer the diagram of Question 39 to an obhque projection and 

show the projecting plane of the hne and the path described by 
the revolving points. 

43. Find where a given line pierces a given plane. Give analysis and 

construction. 

44. Transfer the diagram of Question 43 to an obhque projection. 

45. Find the distance of a given point from a given plane. Give analysis 

and construction. 

46. Transfer the diagram of Question 45 to an oblique projection. Omit 

the portion of the construction requiring the revolution of the 
points. 

47. Find the distance from a given point to a given line. Give analysis 

and construction. 

48. Transfer the diagram of Question 47 to an oblique projection. Omit 

the portion of the construction requiring the revolution of tlie 
points. 

49. Find the angle between two given intersecting hncs. Give analysis 

and construction. 

50. Make an ()l)h(iue projection of the diagram in Question 49. 

51. Find the angle between two given planes. Give analysis and con- 

struction. 

52. Make an oblicjue projection of the diagram in (^)ues(i()n 51. 



136 GEOMETRICAL PROBLEMS IN PROJECTION 

53. Find the angle between a given plane and the horizontal plane of 

projection. Give analysis and construction. 

54. Transfer the diagram of Question 53 to an oblique projection. 

55. Find the angle between a given plane and the vertical plane of pro- 

jection. Give analysis and construction. 

56. Transfer the diagram of Question 55 to an oblique projection. 

57. Draw a plane parallel to a given plane at a given distance from it. 

Give analysis and construction. 

58. Project a given line on a given plane. Give analysis and construction. 

59. Make an obhque projection of the diagram in Question 58. 

60. Find the angle between a given hne and a given plane. Give analysis 

and construction. 

61. Make an oblique projection of the diagram in Question 60. 

62. Find the shortest distance between a pair of skew lines. Give 

analysis and construction. 

63. Make an oblique projection of the diagram in Question 62. 

64. Through a given point, draw a Une of a given length, making given 

angles with the planes of projection. 

65. Show the construction for the three remaining cases of the problem. 

in Question 64. 

66. Through a given point, draw a plane, making given angles with the 

principal planes. 

67. Prove that the construction in Question 66 is correct by finding 

the angles that the given plane makes with the principal planes. 
(Note that there are four possible cases of this problem.) 

68. Show the construction for the three remaining cases of the problem 

in Question 66. 

69. Prove that the constructions in Question 68 are correct by finding 

the angles that the given plane makes with the principal planes. 

70. Through a given Hne, in a given plane, draw another hne intersecting 

it at a given point, and at a given angle. 

71. Transfer the diagram of Question 70 to an obhque projection. 

72. Through a given hne in a given plane, pass another plane, making^ 

given angles with the given plane. 

73. Make an obhque projection of the diagram in Question 72. 

74. Construct the projections of a circle lying in a given obhque plane,. 

the diameter and its centre in the given plane being known. 

Note. The following exercises embrace operations in all four 
angles. 

75. Given a hne the first angle and a point in the second angle, draw 

a hne through the given point, parallel to the given line. 

76. Given a hne in the first angle and a point in the third angle, draw 

a hne through the given point, parallel to the given line. 

77. Given a hne in the first angle and a point in the fourth angle, draw 

a hne through the given point, parallel to the given hne, 

78. Given a line in the second angle and a point in the third angle, 

draw a hne through the given point, parallel to the given line. 



THE POINT, THE LINE, AND THE PLANE 137 

79. Given a line in the third angle and a point in the fourth angle, draw 

a hne through the given point, parallel to the given hne. 

80. Given a Hne in the fourth angle and a point in the second angle, 

draw a hne through the given point, parallel to the given hne. 
8L Draw two intersecting hues in the second angle . 

82. Draw two intersecting lines in the third angle. 

83. Draw two intersecting lines in the fourth angle. 

84. Find where a given line pierces the principal planes when the hmited 

portion of the hne is in the second angle. 

85. Find where a given line pierces the principal planes when the limited 

portion of the hne is in the third angle. 

86. Find where a given hne pierces the principal planes when the limited 

portion of the line is in the fourth angle. 

87. Show the second angle traces of a plane passed through a second 

angle oblique line. 

88. Show the third angle traces of a plane passed through a third angle 

obhque hne. 

89. Show the fourth angle traces of a plane passed through a fourth 

. angle oblique line. 

90. Given a hne in the second angle and parallel to the ground line, 

pass an oblique plane through it. Use a profile plane as part of 
the construction. 
9L Given a hne in the third angle and parallel to the ground Une, 
pass an oblique plane through it. Use a profile plane as a part 
of the construction. 

92. Given a hne in the fourth angle and parallel to the ground hne, 

pass an oblique plane through it. Use a profile plane as a part 
of the construction. 

93. Given a point in the second angle, pass an oblique plane through 

it. Show only second angle traces. 

94. Given a point in the third angle, pass an obhque plane through it. 

Show only third angle traces. 

95. Given a point in the fourth angle, pass an obhque plane through 

it. Show only fourth angle traces. 

96. Given the second angle traces of two obhque planes, find their 

intersection in the second angle. 

97. Given the tliird angle traces of two obhque planes, find their inter- 

section in the third angle. 

98. Given the fourth angle traces of two oblique planes, find tluir 

intersection in the fourth angle. 

99. Given the first angle traces of an obli(]ue plane and one ])roje('tion 

of a second angle point, find the corresponding projection. 

100. Given the first angle traces of an oblique piano, and one projection 

of a third angle point, find the corresiwnding projection. 

101. Given the first angle traces of an obli(iU(^ plane, and one i)roje('tion 

of a fourth angle ]K)int, find the corresponding projection. 

102. Given the second angle traces of an obli(iue i)lane, and one projec- 

tion of a third angle point, find the corresponding projection. 



138 GEOMETRICAL PROBLEMS IN PROJECTION 

103. Given the third angle traces of an oblique plane, and one pro- 

jection of a fourth angle point, j&nd the corresponding pro- 
jection. 

104. Given the first angle traces of an obhque plane, and a point in the 

second angle, pass a plane through the given point and parallel 
to the given plane. 

105. Given the first angle traces of an obhque plane, and a point in the 

third angle, pass a plane through the given point and parallel to 
the given plane. 

106. Given the first angle traces of an oblique plane, and a point in the 

fourth angle, pass a plane through the given point and parallel 
to the given plane. 

107. Given the second angle traces of an oblique plane, and a point in 

the third angle, pass a plane through the given point and parallel 
to the given plane. 

108. Given the third angle traces of an obhque plane, and a point in 

the fourth angle, pass a plane through the given point and parallel 
to the given plane. 

109. Given the first angle traces of an oblique plane and a point in the 

second angle, draw a line through the given point perpendicular 
to the given plane. 

110. Given the first angle traces of an obhque plane and a point in the 

third angle, draw a line through the given point perpendicular 
to the given plane. 

111. Given the first angle traces of an obhque plane and a point in the 

fourth angle, draw a line through the given point perpendicular 
to the given plane. 

112. Given the second angle traces of an oblique plane and a point in 

the fourth angle, draw a line through the given point perpendicular 
to the given plane. 

113. Given the first angle traces of a plane perpendicular to the hori- 

zontal plane but inchned to the vertical plane, and a point in 
the third angle, draw a line through the given point perpendicular 
to the given plane. 

114. Given the third angle traces of a plane perpendicular to the vertical 

plane but inclined to the horizontal plane, and a point in the 
fourth angle, draw a line through the given point, perpendicular 
to the given plane. 

115. Given the first angle traces of a plane parallel to the ground hne 

and a point in the third angle, draw a line through the given point 
perpendicular to the given plane. 

116. Draw a profile plane of the diagram in Question 115. 

117. Make an oblique projection of the diagram in Question 115. 

118. Given the fourth angle traces of a plane parallel to the ground hne 

and a point in the second angle, draw a hne through the given 
point perpendicular to the given plane. 

119. Draw a profile plane of the diagram in Question 118. 

120. Make an oblique projection of the diagram in Question 118. 



THE POINT, THE LINE, AND THE PLANE 139 

121. Given a line in the first angle and a point in the second angle, pass 

a plane through the given point perpendicular to the given 
line. 

122. Given a line in the first angle and a point in the third angle, pass a 

plane through the given point perpendicular to the given line. 

123. Given a Hne in the first angle and a point in the fourth angle, pass 

a plane through the given point perpendicular to the given line. 

124. Given a line in the second angle and a point in the third angle, 

pass a plane through the given point perpendicular to the given 
line. 

125. Given a line in the third angle and a point in the fourth angle, pass 

a plane through the given point perpendicular to the given hne. 

126. Given three points in the second angle, pass a plane through them. 

127. Given three points in the third angle, pass a plane through them. 

128. Given three points in the fourth angle, pass a plane through them. 

129. Given two points in the first angle and one point in the second 

angle, pass a plane through them. 

130. Given a point in the first angle, one in the second angle, and one 

in the third angle, pass a plane through them. 

131. Given a point in the second angle, one in the third angle, and one 

in the fourth angle, pass a plane through them. 

132. Given a hne in the horizontal plane and a point in the second angle, 

revolve the point about the line until it coincides with the hori- 
zontal plane. 

133. Given a line in the horizontal plane and a point in the third angle, 

revolve the point about the hne until it coincides with the hori- 
zontal plane. 

134. Given a line in the horizontal plane and a point in the fourth angle 

revolve the point about the hne until it coincides with the hori- 
zontal plane. 

135. Given a hne in the vertical plane and a point in the second angle, 

revolve the point about the hne until it coincides with the vertical 
plane. 

136. Given a hne in the vertical plane and a point in the fourth angle, 

revolve the point about the hne until it coincides with the vertical 
plane. 

137. Given two points in the second angle, find the true distance between 

them. 

138. Given two points in the third angle, find the true distance betwecMi 

them. 

139. Given two points in the fourth angle, find the true distance between 

them. 

140. Given one point in the first angle and one jwint in the second angle, 

find tlu^ true distance betwcHMi them, 

141. Given one point in the first angle and one i)()int in the third angl(\ 

find the truc^ distance between them. 

142. Given one point in the second angle and one point in the tliird 

angle, find the true distance betwcHMi them. 



140 GEOMETRICAL PROBLEMS IN PROJECTION 

143. Given one point in the second angle and one point in the fourth 

angle, find the true distance between them. 

144. Given the first angle traces of a plane and a line in the second angle, 

find where the given line pierces the given plane. 

145. Given the first angle traces of a plane and a fine in the third angle, 

find where the given line pierces the given plane. 
140. Given the first angle traces of a plane and a line in the fotirth angle, 

find where the given line pierces the given plane. 
147. Given the second angle traces of a plane and a fine in the third 

angle, find where the given fine pierces the given plane. 
14S. Given the second angle traces of a plane and a line in the fourth 

angle, find where the given line pierces the given plane. 

149. Given the tliird angle traces of a plane and a line in the fourth 

angle, find where the given line pierces the given plane. 

150. Given the first angle traces of a plane and a point in the second 

angle, find the distance from the point to the plane. 

151. Given the first angle traces of a plane and a point in the third angle, 

find the distance from the point to the plane. 

152. Given the first angle traces of a plane and a point in the fourth 

angle, find the distance from the pomt to the plane. 

153. Given the second angle traces of a plane and a point in the third 

angle, find the distance from the point to the plane. 

154. Given the second angle traces of a plane and a point in the fourth 

angle, find the distance form the point to the plane. 

155. Given the third angle traces of a plane and a point in the fourth 

angle, find the distance from the point to the plane. 

156. Given a line in the first angle and a point in the second angle, find 

the distance from the point to the line. 

157. Given a hue in the first angle and a point in the third angle, find 

the distance from the point to the line. 
15S. Given a fine in the first angle and a point in the fourth angle, find 
the distance from the point to the line. 

159. Given a line m the second angle and a pomt in the third angle, 

find the distance from the point to the line. 

160. Given a fine in the second angle and a point in the fourth angle, 

find the distance from the point to the line. 

161. Given a line in the tliird angle and a point in the fourth angle, find 

the distance from the point to the line. 

162. Given two intersecting lines in the second angle, find the angle 

between them. 

163. Given two intersecting lines in the third angle, find the angle between 

them. 

164. Given two mtersecting lines in the fourth angle, find the angle 

between them. 

165. Given two intersecting lines, one in the first and one in the second 

angle, find the angle between them. 

166. Given two intersecting lines, one in the first and one in the third 

angle, find the angle between them. 




THE POINT, THE LINE, AND THE PLANE 141 

167. Given two intersecting lines, one in the first and one in the fourth 

angle, fiind the angle between them. 

168. Given two intersecting lines, one in the second and one in the third 

angle, find the angle between them. 

169. Given two intersecting lines, one in the second and one in the fourth 

angle, find the angle between them. 

170. Given two intersecting fines, one in the third and one in the fourth 

angle, find the angle between them. 
17L Given the second angle traces of two intersecting planes, find the 
angle between them. 

172. Given the third angle traces of two intersecting planes, find the 

angle between them. 

173. Given the fourth angle traces of two intersecting planes, find the 

angle between them. 

174. Given the first angle traces of one plane and the second angle traces 

of another plane, find the angle between them. 

175. Given the first angle traces of one plane and the third angle traces 

of another plane, find the angle between them. 

176. Given the first angle traces of one plane and the fourth angle traces 

of another plane, find the angle between them. 

177. Given the second angle traces of one plane and the third angle 

traces of another plane, find the angle between them. 

178. Given the second angle traces of one plane and the fourth angle 

traces of another plane, find the angle between them. 

179. Given the third angle traces of one plane and the fourth angle traces 

of another plane, find the angle between them. 

180. Given a plane in the second angle, find the angle between the given 

plane and the horizontal plane. 

181. Given a plane in the second angle, find the angle between the given 

plane and the vertical plane. 

182. Given a plane in the third angle, find the angle between the given 

plane and the horizontal plane. 

183. Given a plane in the third angle, find the angle between the given 

plane and the vertical plane. 

184. Given a plane in the fourth angle, find the angle between the given 

plane and the horizontal i)lune. 

185. CJiven a plane in the fourtli angle, find the angle between the given 

plane and the vertical jihuie. 

186. Given the second angle traces of a plane, draw another parallel 

plane at a given distance from it. 

187. Given the third angle traces of a plane, draw another i)arallel plane 

at a given distance from it. 

188. Given the fourth angle traces of a i)lane, draw anotlu^r j^arallel 

plane at a given distanci^ from it, 

189. (liven the first angle traces of a jilane and a line in the second angle, 

l^roject the given line on the given plane. 

190. (liven the first angle traces of a phxnv and a line in the third angle, 

project the giv(>n liiu^ on {hv gixcn plan(\ 



142 GEOMETRICAL PROBLEMS IN PROJECTION 

19L Given the first angle traces of a plane and a line in the fourth angle, 
project the given line on the given plane. 

192. CTiven the second angle traces of a plane and a line in the third angle, 

project the given line on the given plane. 

193. Given the second angle traces of a plane and a line in the fourth 

angle, project the given line on the given plane. 

194. Ciiven the third angle traces of a plane and a hne in the fourth angle, 

project the given hne on the given plane. 

195. Given the fouith angle traces of a plane and a hne in the second 

angle, project the given line on the given plane. 

196. Cnven the first angle traces of a plane and a hne in the second angle, 

find the angle between the given hne and the given plane. 

197. Given the fii'st angle traces of a plane and a line in the third angle, 

find the angle between the given line and the given plane. 

19S. Given the first angle traces of a plane and a line in the fourth angle, 

find The angle between the given line and the given plane. 

199. Criven the second angle traces of a plane and a line in the third angle, 

find the angle between the given line and the given plane. 

200. CtIvcu the second angle traces of a plane and a line in the fourth 

angle, find the angle between the given line and the given plane 

201. Cxiven the third angle traces of a plane and a line in the fourth angle, 

find the angle between the given line and the given plane. 

202. CTiven the fourth angle traces of a plane and a hne in the second 

angle, find the angle between the given hne and the given plane. 

203. Given the fourth angle traces of a plane and a line in the third angle, 

find the angle between the given line and the given plane. 

204. Cxiven a hne in the first angle and another hne in the second angle 

(skew lines), find the shortest distance between them. 

205. Given a line in the first angle and another line in the third angle 

(skew lines), find the shortest distance between them. 

206. Given a line in the first angle and another line in the fourth angle 

(skew lines), find the shortest distance between them. 

207. Given a hne in the second angle and another line in the third angle 

(skew lines), find the shortest chtstance between them. 

208. Given a line in the second angle and another line in the fourth 

angle (skew fines) ; find the shortest distance between them. 

209. Given a line in the third angle and another line in the fomth angle 

(skew hnes), find the shortest distance between them. 

210. Through a given point in the second angle, draw a hne of a given 

length, making given angles with the planes of projection. 

211. Through a given point in the third angle, draw a hne of a given 

length, making given angles with the planes of projection. 

212. Tlirough a given point in the fourth angle, draw a line of a given 

length, making given angles with the planes of projection. 

213. Through a given point in the second angle, draw a plane making 

given angles with the principal planes. 

214. Through a given point in the third angle, draw a plane making 

given angles vdxh the principal planes. 



THE POINT, THE LINE, AND THE PLANE 143 

215. Through a given point in the fourth angle, draw a plane making 

given angles with the principal planes. 

216. Given the second angle traces of a plane, a line, and a point in the 

plane, draw another Hne through the given point, making a 
given angle with the given hne. 

217. Given the third angle traces of a plane, a line, and a point in the 

plane, draw another hne through the given point, making a 
given angle with the given hne. 

218. Given the fourth angle traces of a plane, a line and a point in the 

plane, draw another line through the given point, making a given 
angle with the given hne. 

219. Given the second angle traces of a plane, and a line in the plane, 

draw through the hne another plane making a given angle with 
the given plane. 

220. Given the third angle traces of a plane, and a line in the plane, 

draw through the hne another plane making a given angle with 
the given plane. 

221. Given the fourth angle traces of a plane, and a hne in the plane, 

draw through the hne another plane making a given angle with 
the given plane. 

222. Given the second angle traces of a plane, the diameter and the 

centre of a circle, construct the projections of the circle. 

223. Given the third angle traces of a plane, the diameter and the centre 

of a circle, construct the projections of the circle. 

224. Given the fourth angle traces of a plane, the diameter, and the 

centre of a circle, construct the projections of the circle. 



CHAPTER IX 

CLASSIFICATION OF LINES 

901. Introductory. Lines are of an infinite variety of forms. 
The frequent occurrence in engineering of certain varieties makes 
it desirable to know their properties as well as their method 
of construction. It must be remembered that lines and points 
are mathematical concepts and that they have no material 
existence. That is to say, a line may have so many feet of length 
but as it has no width or thickness, its volume, therefore, is 
zero. Hence, it cannot exist except in the imagination. Like- 
wise, a point is a still further reduction and has position only; 
it has no dimensions at all. Of course, the material representa- 
tion of lines and points requires finite dimensions, but when 
speaking of them, or representing them, it is the associated 
idea, rather than the representation, which is desired. 

902. Straight line. A straight line may be defined as the 
shortest distance between two points.* It may also be described 
as the locus (or path) of a generating point which moves in the 
same direction. Hence, a straight line is fixed in space by two 
points, or, by a point and a direction. 

903. Singly curved line. A singly curved line is the locus 

(plural :-loci) of a generating point which moves in a varying 
direction but remains in a single plane. Sometimes the singly 
curved fine is called a plane curve because all points on the 
curve must he in the same plane. 

904. Representation of straight and singly curved lines. 

Straight and singly curved lines are represented by their pro- 
jections. When singly curved lines are parallel to the plane 

* Frequently, this is called a right line. There seems no reason, however, 
why this new nomenclature should be used; hence, it is here avoided. 

144 



CLASSIFICATION OF LINES 145 

of projection, they are projected in their true form and require 
only one plane of projection for their complete representation. 
If the plane of the curve is perpendicular to the principal planes, 
a profile will suffice. If the plane of the curve is inclined to the 
planes of projection, both horizontal and vertical projections 
may be necessary, unless a supplementary plane be used, which 
is parallel to the plane of the curve.* This latter condition is 
then similar to that obtained when the plane of the curve is 
parallel to one of the principal planes. 

905. Circle. The circle is a plane curve, every point of which 
is equidistant from a fixed point called the centre. The path 
described by the moving point (or locus) is frequently designated 
as the ''circumference of the circle."! To define such curves 
consistently, it is necessary to limit the definition to the nature 
of the line forming the curve. Therefore, what is usually known 
as the circumference of the circle should simply be known as 
the circle and then subsequent definitions of other curves become 
consistently aUke. 

In Fig. 144 a circle is shown; abedc is the curve or circle 
proper. The fixed point o is called the centre 
and every point on the circle is equidistant from 
it. This fixed distance is called the radius; oa, 
DC, and ob are all radii of the circle; be is a 
diameter and is a straight line through the 
centre and equal to two radii in length. The 
straight line, de, limited by the circle is a chord 
and when it passes through the centre it 
becomes a diameter; when the same line is extended like gh it is 
a secant. A limited portion of the circle like ac is an arc; when 
equal to one-quarter of the whole circle it is a quadrant; when 
equal to one-half of the whole circle it is a semicircle as cab 
or bfc. The area included between two radii and the circle is 
a sector as aoc; that between any line like de and the circle is 

* In all these cjisos, the plane of the curve may be made to coineiile with 
the plane of projection. This, of course, is a special case. 

t The introduction of this term makes it necessary to define the circle 
as the enclosed area. This condition is unfortunate, as in the case of the 
parabola, hyperbola, and numerous other curves, the curves are open and do 
not enclose an area. 




146 



GEOMETRICAL FROBLEMS IX PROJECTION 



a segment. If any other circle is dra^vn with the same centre o 
the circles are concentric, otherwise eccentric; the distance 
betw^een the centres in the latter case is the eccentricity. 

906. Ellipse. The ellipse is a plane curve, in which the 
smn of the distances of any point on the curve from two fixed 
points is constant. Fig. 145 shows this curve as adbc; The 
major axis is ab, and its length is the constant distance of the 
definition; cd is the minor axis and is perpendicular to ab through 
its middle point o. The fixed points or foci (either one is a focus) 
are e and f and are located on the major axis. 

If ab be assumed as the constant distance, and e and f be 
assumed as the foci, the minor axis is determined by drawing 
arcs from e and f with oa, equal to one-half of the major axis, 





Fig. 146. 



as a radius. Thus ec-|-cf = ab and also ed+df = ab. To find 
any other point on the curve, assume any distance as bg, and 
with bg as a radius and f as a centre, draw an arc fh. With 
the balance of the major axis ag as a radius, draw an arc eh 
from the focus e as a centre. These intersecting arcs locate h, 
a point on the curve. Thus eh+hf = ab and therefore satisfies 
the definition of the curve. With the same radii just used, the 
three other points i, j and k are located. In general, four points 
are determined for any assumption except for the points a, c, b 
or d. It can be shown that a and b are points on the curve, 
because oa = ob and oe = of. Hence ae=fb, therefore, fa-|-ea = 
ab and also eb+fb=ab. 

In the construction of this, or any other curve, the student 
should avoid trj-ing to save time by locating only a few points. 
This is a mistaken idea, as, \^dthin reasonable limits, time is 
saved by dra^vang numerous points on the curve, particularly 



CLASSIFICATION OF LINES 



147 



where the curve changes its direction rapidly. The direction of 
the curve at any point should be known with a reasonable degree 
of accuracy. 

Another method of drawing an elhpse is shown in Fig. 146. 
It is known as the trammel method. Take any straight ruler 
and make oc = a and od = b. By locating c on the minor axis 
and d on the major axis, a point is located at o, as shown. This 
method is a very rapid one, and is the one generally used when 
a true ellipse is to be plotted, on account of the very few lines 
required in the construction. Both methods mentioned are 
theoretically accurate, but the latter method is perhaps used 
oftener than the former. 

In practice, ellipses are usually approximated by employing 
four circular arcs, of two different radii as indicated in Art. 405. 
The major and minor axes are laid off and a smooth looking 
curve drawn between these limits. Of course, the circular arcs 
do not produce a true ellipse, but as a rule, this method is a 
rapid one and answers the purpose in conveying the idea. 



907. Parabola. The parabola is a plane ciirve, which is 
the locus of a point, moving so that the distance from a fixed 
point is always equal to the distance 

from a fixed line. The fixed point is the 
focus and the fixed line is the directrix. 
A line through the focus and perpen- 
dicular to the directrix as eg, Fig. 147, 
is the axis with respect to which the curve 
is symmetrical. The intersection of the 
axis and the curve is the vertex shown 
atf. 

The point a on the curve is so situated 
that ac = ad; also b is so situated that 
be = be. Points beyond b become more and 

more remote, from both the axis and directrix. Hence, it is 
an open curve extending to infinity. Discussions are usually 
limited to some finite portion of the curve. 

908. Hyperbola. The hjrperbola is a plane curve, traced 
by a point, which moves so that the difference of its distances 
from two fixed points is constant. The fixed points a and b, 




Fig. 147. 



148 



GEuMETPvICAL PROBLEM:^ IX PROJECTION 




Pig. 1-iS. 



Fig. 14S. are the foci. The Hne ab, passing through them, is 
the transTerse * axis ; the point at which either curve crosses 
the axis as e or f is the vertex > plural: — vertices), and the line kl, 
perpendicular to ab at its middle point, is the conjugate axis. 
To draw the ciir\-e. lay off the foci a and b; also lay off ef, 

the constant cLLstance., so that 
eo = of and o is the middle 
of ab. It must be observed 
that ef is always smaller than 
ab otherwise the cin^*e cannot 
be constructed. Take any 
radius be, greater than bf, 
and draw an indefinite arc; 
from a, draw ac, so that ac 
— bc = ef, hence. ac=bc+ef. 
The point c is thus on the 
cur^e. Similarly, draw an 
arc ad, from a, and another 
arcbd = ad-ref. This locates 
d, which, in this case, is on 
another ctirve. In spection will show that there are two 
branches of this curve. The point c has been selected so as to 
be on one branch, and d, on the other. To construct the curve, 
accurately, many more points must be located than shown. 
Both branches are open and symmetrical with respect to both 
axes, 

A tangent to the curve through o is called an asymptote 
when it touches the h}-perbola in two points, each at an infinite 
distance from o. As will be observed, there are two as^Taptotes. 

909. Cycloid. The cycloid is a plane curve, traced by a 
point on a circle which rolls over a straight line. The straight 
line over which the circle roUs is the directrix; the point on the 
circle may be considered as the generating point. 

The curve is shown as abc ... i in Fig. 149. To construct 
it. lay out the directrix ai, and to one side, draw an auxiliary 
circle equal in diameter to the roUing circle as shown at 1' 2', 
etc. On a line through the centre of the auxiliai^* circle, draw 
the line 3-9 parallel to the directrix ai. Lay off the distance 



* Sometimes known iis the "•principal axis. 



CLASSIFICATION OF LINES 



149 



1-9 on this line, equal to the length of the circle (3.14 X diameter). 
Assume that when the centre of the rolling circle is at 1, a is 
the position of the generating point. After one-eighth of a 
revolution the centre has moved to 2, where the distance 1-2 
is one-eighth of the distance 1-9. The corresponding position 
of the generating point is shown at 2' in the auxiliary circle at 
the left. Hence, for one-eighth of a revolution the centre of the 
rolling circle has moved from 1 to 2 and the generating point 
has moved a distance vertically upward equal to the distance 
of 2' above the line ai. Therefore, draw a line through 2', parallel 
to ai; and then from 2 as a centre, draw an arc with the radius 
of the rolling circle intersecting this line in the point b, a point 
on the required curve. After a quarter of a revolution, the gen- 
erating point is above the directrix, at a height equal to the 




Fig. 149. 

distance of 3' above ai. It is also on an arc from 3, with a radius 
equal again to the radius of the rolling circle; hence, c is the point. 
This process is continued until the generating point reaches its 
maximum height e after one-half of a revolution, when it begins 
to descend, to i, as shown, after completing one revolution. 
Further rolling of the circle causes the generating point to dupli- 
cate its former steps, as it continues along the directrix to infinity 
if desired. 

The cycloid is the same curve that is produced by a mark 
on the rim of a car-wheel, while rolling along the track, only, 
here, no slipping is permitted. The cycloid is, thus, continuous, 
each arch being an exact duplicate of the preceding. 



910. Epicycloid. The epicyloid is a plane curve, which is 
generated by a point on a circle which rolls on the outside of 
another circle. The directrix is hero an arc of a circle, instead 
of a straight hnc. 



150 



GEOMETRICAL PROBLEMS IN PROJECTION 



The construction is indicated in Fig. 150. The centre of the 
rolling circle assumes successive positions as 1, 2, 3, etc. The 
length of the arc ai is equal to the length of the rolling circle. 
The points are located much the same as in the cycloid, and, as 




the necessary construction lines are shoTsu, the student should 
have no difficulty in following the construction. 

911. Hypocycloid. The hjf^ocycloid is a plane cixrve/ 
generated by a point on a circle which rolls on the inside o 




Fig. 151. 



another circle. The directrix of the epicycloid and the hj-pocy- 
cloid may be the same, the epicycloid is described on the outside 
of the arc while the hj-pocycloid is described on the inside. 

All the necessary construction lines required to draw the 
hj-pocycloid have been added in Fig. 151 and the method is 



CLASSIFICATION OF LINES 151 

perhaps evident. No confusion should arise even though the 
curves are drawn in positions other than those shown. For 
instance, the cycloids (this includes the cycloids, epicycloid and 
hypocycloid) could be shown upside-down; the curves are gener- 
ated in much the same way and their properties, therefore, do 
not change. 

912. Spiral. The spiral is a plane curve, generated by a 
point moving along a given line while the given line is revolving 
about some point on the line. An infinite number of spirals 
may exist, because the point may have a variable velocity along 
the line, while again the line may have a variable angular velocity 
about the point. The one having uniform motion of the point 
along the line, and uniform angular velocity about the point 
is called the Spiral of Archimedes. 

Fig. 152 shows the Archimedean spiral which is perhaps 
the simplest type of spiral. If ox be 
assumed as the primitive position of a line 
revolving at o, and the point also starts at 
o, then o is the starting point of the curve. 
Suppose that after one-eighth of a revo- 
lution the generating point has moved a 
distance oa, then, after one-quarter of a 
revolution, the point will be at b, where 
ob = 2Xoa, and soon. The spiral becomes Yig. 152. 

larger and larger as the revolution con- 
tinues. Had the line revolved in the other direction, the curve 
would have been the reverse of the one shown. 

913. Doubly curved line. A doubly curved line is one 
whose direction is continually changing and whose points do 
not lie in one plane. A piece of wire may be twisted so as to 
furnish a good example of a doubly curved line. 

914. Representation of doubly curved lines. As no 

single plane can contain a doubly curved line, it becomes neces- 
sary to use two planes and project, orthographically, the line 
on them. Familiarity is had with the representation of points 
in space on the i^rincipal ])lanes. Thc^ line may be conceived as 
being made up of an infinite number of i^oints and each ])oint 
can be located in space by its projections. 




152 



GEOMETRICAL PROBLEMS IX PROJECTION 




Fig. 153. 



Fig. 153 represents a curve ABCD, shown by its horizontal 
projection abed and its vertical projection a'b'c'd'. Any point 

on the curve, such as B, is found 
by erecting perpendiculars from b 
and b' and extending them to their 
intersection; this will be the point 
sought. The principal planes must 
be at right angles to each other if 
it is desired to locate a point by 
erecting these perpendiculars; other- 
wise, the curve must be imagined 
from its projections when the planes are revolved into coinci- 
dence, as is customary in orthographic projection. 

915. Helix. The helix is a doubly curved line described 
by a point having motion around a line called the axis, and in 
addition, a motion along it. Unless it is noted otherwise, the 
helix will be considered as having a uniform circular motion 
around the axis and also a uniform motion along it. The curve 
finds its most extensive application on that t;>T3e of screw known 
as a machine screw. This is then a uniform cylindrical helix. 
Wood screws furnish examples of conical helices.* The helix 
is also frequently used in making springs of a type known as 
helical springs. f 

Fig. 154 shows the construction of the helix. Assume that 
the drawing is made in the third angle of projection; the plan is 
therefore on top. The ground line is omitted because the dis- 
tance of the points from the principal planes is not required, 
but only their relative location to each other. To return, 1', 
2', . . . 8' is the plan (horizontal projection) of the helix, showing 
the circular motion of the point around the axis ab. In the 
elevation (vertical projection), the starting point of the curve is 
shown at 1. If the point revolves in the direction, of the arrow, 
it will, on making one-eighth of a revolution, be at 2' in plan 



* The distinction might be made between cylindrical and conical helices 
by considering the cm-ve as being drawn on the surface of a cylinder or a cone 
as the case may be. 

t Helical springs are frequently although incorrectly called spiral springs. 
The spiral has been previously defined; and a spring of that shape is a spiral 
spring. 



CLASSIFICATION OF LINES 



153 



and at 2 in elevation. After a quarter of a revolution, the point 
is at 3' in plan, and at 3 in elevation. The position 3 is its extreme 
movement to the right, for at 4, the point has moved to the left, 
although continually upward as shown in the elevation. On 
completion of one revolution, the point is at 9, ready to proceed 
with an identical curve beyond it. 

The distance between any point and the position of the point 
after one complete revolution is known as the pitch. The dis- 
tance p is that pitch, and may be given from any point on 
the curve to the succeeding position of that point after one 



'' 


6 

( 


h 


5' 

7 


^ 


3' 


% 


V- 










/ 










^ 










_,rf^ 
































^ 














^ 














■v 












^ 


,3 








^ 


^ 













7' 








> 




^i- 










'^ 


•i 












^ 


4 












^ 








>' ^ 








1 


^ 










a 


r 




Fig. 154. 



Fig. 155. 



revolution. The distance 01', 02', etc. is the radius of the 
helix. 

Fig. 155 is a double helix and consists simply of two distinct 
helices, generated so that the starting point of one helix is just 
one-half of a revolution ahead or behind the other helix. The 
portion of the helix that is in front of the axis is shown in full; 
that behind the axis is shown dotted to give the effect of a helix 
drawn on a cylindcT. The pitch again is measured on any one 
curve and requires a complete revolution. It must not be 
measured from one curve to a similar position on tlu^ next curve 
as the two helices are entirely distinct. Fig. 155 will show this 
correctly. Double, and even triple and quadruple helices are 
used on screws; they are simply interwoven so as to be equal 
(listniices apart. 



154 



GEOMETRICAL PROBLEMS IX PROJECTION 



916. Classification of lines. 



Lines 



Straight Lines: UnidirectionaL 



Curved Lines : 



Singly Curved Lines: Di- 
rection continually chang- 
ing, but always in a single 
plane. 



circle 

ellipse 

parabola 

hjTDerbola 

cj'cloid 

epicycloid 

hypocycloid 

spiral 

etc. 



Doubly Curved Lines: Di- 
rection continually chang- I helix 
ing, but not in any one 1 etc. 
plane. 




Fig. 156. 



917. Tangent. A tangent is the limiting position of a 
secant as the points of secancy approach and ultimately reach 
coincidence. Suppose, in Fig. 156, pq is a secant to the curve 

and that it revolves about p as a 
centre. At some time, the secant 
pq will assume some such position as 
pr; the point q has then moved to r 
and if it continues, it will pass 
through p and reach s. The secant 
then intersects on the opposite side 
of p. If the rotation be such that 
q passes through r and ultimately 
coincides vrith. p, then this limiting 
position of the secant, sho^Ti as pt, is the tangent to the curve 
and p is the point of tangency. It will be observed that the 
tangent is fixed by the point p, and the direction of the limiting 
position of the secant. 

The condition of tangency is a mutual relation. That is, 
the curve is tangent to the line or the line is tangent to the curve. 
Also, two or more curves may be tangent to each other because 
the tangent line may be considered (at the point of tangency) 
as the direction of the curve (see Art. 920). 

918. Construction of a tangent. If a tangent is to be 
drawn to a curve from an outside point, the drafting room method 
is to use a ruler of some sort and place it a slight distance away 
from the point and then revolve it until it nearly touches the 



CLASSIFICATION OF LINES 155 

curve; a straight line then drawn through the point and touching 
the curve will be the required tangent. 

If the problem is to draw a tangent at a given point on a 
curve, however, the quick method would be to estimate the 
direction so that the tangent appears to coincide with as much 
of the curve on one side as it does on the other. 

A more accurate method of constructing the tangent at a 
given point on a curve is shown in Fig. 157. Let a be the desired 
point of tangency. Draw through a, the secants ab, ac, ad, 
the number depending upon the degree of accuracy, but always 
more than here shown. With a as a centre, draw any indefinite 
arc eh, cutting the prolongations of the secants. Lay off the 
chord ab = ei, ac = fj, and ad=hk, but this latter chord is laid 




Fig. 157. 

off to the left of the arc, because its secant cuts the curve to the 
left of the desired point of tangency. A smooth curve may 
now be drawn through i, j, and k, and where this curve inter- 
sects the indefinite arc at g, draw ga, the required tangent. The 
proof of this is quite simple. If ab, ac and ad be considered as 
displacements of the points b, c and d from the positions that 
they should occupy when the secant becomes a tangent, then, 
when these points approach a, so as ultimately to coincide with 
it, the displacement is zero. The secant then becomes a tangent. 
The curve ijk is a curve of displacements from efh, the inch^finite 
arc. Hence, the desired tangent must pass through g, as it 
lies on the curve efh and its displacement from that curve is 
therefore zero. 

919. To find the point of tangency. On the other hand, 
if the tangent is (h'awu, and it is desired to find the point of 



156 



GEOMETRICAL PROBLEMS IN PROJECTION 





5 

\ 


y 


o <^ 


\ 


i" 


c 


W-^:-/ 


T 


i k t\ "' 

1 


« T 

\ 
r* 



Fig. 158. 



tangency, the problem becomes slightly different. Suppose TT, 
Fig. 158, is the given tangent, and atb is the given curve. It is 
desired to find the point of tangency t.. Draw ab, cd and ef, 

any chords parallel to TT. Lay off 
hg, through a, perpendicular to TT, 
and br, through b, perpendicular 
also to TT. Makegh = qr = ab. In 
the other cases, make ij=op = cd, 
and kl = mn = ef. Now draw a 
smooth curve through hjlnpr. 
WTiere this curve intersects TT at 
t, the desired point of tangency is 
found. Again, the proof is quite 
simple. The lengths of the vertical 
lines above and below TT are equal 
to the lengths of the corresponding 
chords. As the chords approach 
the tangent, they diminish in length and ultimately become 
zero. Wliere the curve crosses the tangent, the chord length is 
zero, and, hence, must be the point of tangenc^^ 

920. Direction of a curve. A curve continually changes 
its direction, but at any given point its direction is along the 
tangent to the curve b}^ definition. It is proved in Mechanics, 
that if forces act on a particle so as to give it a curved motion, 
the particle ^-ill fly off along a tangent when the impressed forces 
cease to act.* 

921. Angle between curves. The angle between two inter- 
secting curves is the same as the angles made by the tangents at 
the point of intersection, because the tangents determine the 
direction of the curve at that point. Hence, to draw smooth 
curves, it is necessary that the tangent at the end of one curve 
should coincide with the tangent at the beginning of the next 
curve. 

922. Intersection of lines. Two lines intersect when they 
have a point in common. 'Wlien the term intersection is used in 
connection with a pair of straight lines, it necessarily implies 
that the lines make an angle Ts-ith each other greater than zero; 



* See Newton's laws of motion in a text -book on Phj'sics. 



CLASSIFICATION OF LINES 157 

since, otherwise, if there is a zero angular relation, the lines become 
coincident and have all points in common. 

When a line meets a curve, the angle between the line and 
the curve at the point of intersection is the same as the angle 
between the line and the tangent to the curve at that point. 
When this angle of intersection becomes zero, however, by 
having the line coincide with the tangent, then it is the special 
case of intersection known as tangency. 

Similarly when two curves intersect their angle of intersection 
is the same as the angle between their tangents at the point of 
intersection. Also, when this angle becomes zero then the curves 
are tangent to each other. 

Thus, in order to differentiate between the two types of 
intersection, angular intersection means intersection at an angle 
greater than zero; and, as a consequence, the intersection at a 
zero angle is known as tangential intersection. When inter- 
section is unmodified, then angular intersection is implied. 

923. Order of contact of tangents. A tangent has 
been previously defined as the limiting position of a secant, as the 
points of secancy approach and ultimately coincide with each 
other. This contact, for simple tangency, is of the first order. 
Two curves may also be taDgent to each other so as to have con- 
tact of the first order, as for example, two circles, internally or 
externally tangent. 

Suppose two curves acb and abe. Fig. 159, have first order 
contact at a, and cut each other at the 
point b. If the curve abe be made to 
revolve about the point a as a centre, 
so as to maintain simple tangency * and 
also to have the point b approach a, then 
at some stage of the revolution the point 
b can be made to coincide with a. 
Under this condition there is tangency Fu;. \')\). 

of a higher order because three points 

were made to ultimately coincide with each other; and it is 
called second order of contact. It is possible to have third 
order of contact with four coincident points; and so on. The 

* In order to make this a rigid domonslratioii, the rontre of curvature 
of the two curves at the point a must not hv tlio same. See Art. 924. 





158 GEO>IETEICAL PKOBUEMS Ds PROJECTION 

order of contact is always one le^ than the number of points 
that approach coincidence. 

924. Osculating circle. Centre off curvature. Let abc. 

Fig. 16C'. r irvc :.i^-i b. a poiiti ihrou^ which a cirde gdbe 

passes. :u :^_ ::ie curve abc in three points d4) and e. If the 

-1 Utt: :: z'lLr circle is properly cho^n, it may be revolved 

about b as a centre so that the points d 

and e will both approach and ultimately 

f coincide with b at the same instant. 

This position of the circle is shown as bf . 

Hence, the circle bf is tangent to the 

^ / curve abc, and is of the seoood order 

Oy^ '^- 'Contact. This circle is the c^coladng 

circle. A- this osculating circle must 

Fig. 160. ni : : t i. t : rc>ach the curvature of 

'i-T ^- T 3.:: _ n finy other circle, its 

radius a: :hr riJi- b is iir raius ;: :_— ^r-e. At every other 

poin: .n i:^ i: ^ ii :t := a new oscuiating circle, of a new 

centre, ai.:* :: .:. i.v~ la:;:^;:^. Thus, the osculating circle is the 

second order tangent circle at the point: an i :j1t radius of 

curvature maybe defined as the radius <:: "_t : iTing circle 

through the point, the centre of c urv at ure rii.^ _t iriitre of the 

osculating circle. 

925. Osculating plane. If a tangent be drawn to any doubly 
curved line, an infinite numba- of planes may be pa^ed so as to 
contain the tangent . If some one position of the plane be sdected 
so as to contain. :ar tan^Ti.: -ni a piercing point of the doubly 
curved line : rr revolution of the plane the 
piercing p«jra: ...i. . z :_>:.^r :: ^pproach and ultimately coincide 
- -a iir : lint of tangency; in this poation, the plane is an 
osc illa tin g plane. 

To put the matter differently, suppose it is deared to find 
the osculating plane at s<jiar ::■: int on a doubly curved hne. In 
this case, draw a pair : — nts to the doubly curved line which 
intersect at the grrea. '^sis.. A plane pased through these 
secants will cut the doubly curved line in three piants. As the 
secants approach tangency. the plane wiU approach osculation, 
and this osculating plane is identical with that of the former 
discussion for the same point on the curve. If the carve under 




CLASSIFICATION OF LINES 159 

consideration be a plane curve, then the secants will lie in the 
plane of the curve, and, hence, the osculating plane of the curve 
will be the plane of the curve. 

926. Point of inflexion. Inflexional tangent. Assume 
a curve dae, Fig. 161, and through some one point a draw the 
secant be. If the secant be revolved about the point a so that 
the points of intersection b and c approach a, at some stage of 
the revolution they will coincide with 

it at the same instant if the point a 

is properly chosen. The point a must 

be such that three points on the curve 

ultimately coincide at the same instant . 

Further than this, the radius of curva- Fi<^- 161. 

ture (centre of the osculating circle) 

must make an abrupt change from one side of the curve to the 

other at this point. The point of inflexion therefore is a point 

at which the radius of curvature changes from one side of the 

curve to the other. The inflexional tangent is the tangent at the 

point of inflexion. It may also be noted that the inflexional 

tangent has a second order of contact (three coincident points) 

and therefore is the osculating line to the curve at the point of 

inflexion. 

927. Normal. A normal to a curve is a perpendicular to 
the tangent at the point of tangency. The principal normal 
lies in the osculating plane. As an infinite number of normals 
may be drawn to the tangent at the point of tangency, the nor- 
mal may revolve about the tangent so as to generate a plane 
which will be perpendicular to the tangent and thus establishes 
a normal plane. In the case of a circle, the radius at the point 
of tangency is normal to the tangent; in such cases, the tangent 
is easily drawn, as it must be perpendicular to the radius at the 
point of contact. 

928. Rectification. When a curve is made to roll on a straight 
line, so that no slip occurs between the curve and the lino, the 
distance measured on the line is equal to the corresponding length 
of the curve. This process of finding the length of the curve is 
called rectification. Commercially applied, the curve is measured 
by taking a divider and stepping off wvy small distances; the 



160 



GEOMETRICAL PROBLEMS IN PROJECTION 



number of steps multiplied by the distance between points will 
give approximately the length of the curve. It must be noticed 
that the divider measures the chord distance, instead of the 
arc distance and is therefore always less than the actual length of 
the curve, but when the distance is taken small enough, the 
accuracy of the final result is proportional to the care taken in 
making the measurement. 




929. Involute and evolute. When a tangent rolls about a 
fixed curve, any point on the tangent describes a second curve 

which is the involute of the first 
curve. Fig. 162 shows this in 
construction. Let aceg be the 
fixed curve, and ab, be the posi- 
tion of a taut string that is wound 
on the curve aceg. If a pencil 
point be attached to the string 
and unwound, the pencil point 
will describe the curve bdfh which 
is the involute of the curve aceg. 
The process is the same as though 
the tangent revolved about the 
curve aceg and some point on the 
tangent acted as the generating point. At a, the radius is ab; 
at c, the radius is cd, which is equal to ab plus the rectified arc 
ac (length of string). It may be observed that the curve aceg 
is the curve of centres for the curve bdfh. 

If the string be lengthened so that ai is the starting position, 
it will describe the curve ijkl which again is an involute. This 
second involute is parallel to the first, because the distance between 
the two curves measured along the rolling tangent (or radius of 
curvature) is constant between the two curves. 

The primitive curve aceg is the evolute. The tangent rolls 
on the evolute and any point on it describes an involute. As 
any point on the tangent will answer as the tracing point, it 
follows that every evolute has an infinite number of involutes, 
all of which are parallel curves. 

Reversing the process of the construction of the involute, the 
method of drawing the primitive curve or evolute is obtained. 
If normals are drawn to the involute, consecutive positions of 




CLASSIFICATION OF LINES 161 

the normals will intersect. The locus of these successive inter- 
sections will regenerate the primitive curve from which they 
have been evolved.* 

Again, the length of the tangent to the e volute is the radius 
of curvature for the involute. As the radius of any circle is 
perpendicular to the tangent at that point, it follows that the 
involute is always normal, point for point, to the evolute, since 
the roUing tangent is the direction of the evolute at the point 
of contact and that again is normal to the involute. 

930. Involute of the circle. The involute of the circle 
is a plane curve, described by a point on a tangent, while the 
tangent revolves about the circle. 

Let o, Fig. 163, be the centre of a circle whose radius is oa. 
Let, also, a be the starting point of 
the involute. Divide the circle in 
any number of parts, always, however, 
more than are shown in the illustra- 
tion. Draw tangents to the various 
radii. On them, lay off the rectified 
arc of the circle between the point of 
tangency and the starting point. For ^. .^^ 

instance, eb equals the rectified length 

of the arc ea; also fc equals the rectified semicircle fea; gd 
equals the rectified length of the arc gfea. The involute may 
be continued indefinitely for an infinite number of revolutions, 
but, discussion is usually centred on some limited portion. 

The curve here shown is approximately the same as that 
described by the end of the thread, when a spool is unwound. 

QUESTIONS ON CHAPTER IX 

1. Why are lines and points considered as mathematical concepts? 

2. How is a straight line defined? 

3. By what two means may a straight line be fixed in space? 

4. What is a generating point? 

5. What is a locus? 

6. What is a singly curved line? 

7. What is a plane curve? 

8. How are phuu) curves represented? 

* The evolute of a circle, therefore, is its centre. 




162 GEOMETRICAL PROBLEMS IN PROJECTION 

9. Show the mode of representing curves when their planes are parallel 
to the plane of projection? When perpendicular? When inchned? 

10. Is it desirable to use the plane of the curve as the plane of projection? 

11. Define the circle. 

12. What is the radius? Diameter? Sector? Segment? 

13. When the chord passes through the centre of the circle, what does 

it become? 

14. What is a secant? Quadrant? Semicircle? 

15. Define concentric circles; eccentric circles; eccentricity. 

16. What is an eUipse? 

17. Define major axis of an elhpse; minor axis; foci. 

18. Describe the accurate method of crawing an elhpse by the intersection 

of circular arcs. 

19. Describe the trammel method of drawing an elhpse. 

20. Draw an eUipse whose major axis is 3" long and whose minor axis 

is 2" long. Use the accurate method of intersecting circular 
arcs. 

21. Construct an ellipse whose major axis is 3" long and whose minor 

axis is H" long. L'se the trammel method. 

22. What is a parabola? 

23. Define focus of a parabola; directrix; axis; vertex. 

24. Is the parabola symmetrical about the axis? 

25. Is the parabola an open or a closed curve? 

26. Construct the parabola whose focus is 2" from the directrix. 

27. What is a hyperbola? 

28. Define foci of hyperbola; transverse axis; conjugate axis; vertex; 

asj^mptote. 

29. How mam^ branches has a hyperbola? Are the}' symmetrical about 

the transverse and conjugate axes? 

30. How many asj^mptotes may be drawn to a hA'perbola? 

31. Is the hyperbola an open or a closed curve? 

32. Construct a hyperbola whose distance between foci is 2" and whose 

constant difference is \" . 

33. What is a cycloid? 

34. Define rolling circle of a cycloid; directrix. 

35. Construct a c^'cloid whose chameter of rolling circle is \\" . Draw 

the curve for one revolution onh^ 

36. What is an epicycloid? 

37. What form of directrix has the epycycloid? 

38. Construct the epicycloid, whose chameter of rolling circle is \\" 

and whose chameter of directrix is 8". Draw the curve for one 
revolution only. 

39. What is a hypocycloid? 

40. What is the form of the directrix of the hypocycloid? 

41. Construct a hypocycloid whose diameter of rolling circle is 2" and 

whose diameter of directrix is 9". Draw the curve for one revolu- 
tion onty. 

42. What is a spiral? 



CLASSIFICATION OF LINES 163 

43. Construct an Archimedian spiral which expands 1|" in a complete 

revolution. Draw the spiral for two revolutions. 

44. What is a doubly curved line? 

45. Draw a doubly curved line with the principal planes in oblique 

projection. 

46. Construct the orthographic projection from Question 45. 

47. What is a hehx? 

48. Define uniform cylindrical helix; conical helix; diameter of helix; 

pitch. 

49. Construct a hehx whose diameter is 2", and whose pitch is 1". 

Draw for two revolutions. 

50. Construct a triple helix whose diameter is 2" and w^hose pitch is S". 

The helices are spaced equally and are to be drawn for 1^ revolu- 
tions. 

51. Make a classification of lines. 

52. What is a tangent? 

53. Is the tangent fixed in space by a point and a direction? 

54. Show that the tangent is the limiting position of a secant. 

55. How is a tangent drawn to a curve from a point outside? 

56. Given a curve and a point on it, draw a tangent by the accurate 

method. Prove that the chord length is zero for the tangent 
position. 

57. Given a curve and a tangent, determine the point of tangency. Prove. 

58. What is the direction of a curve? 

59. What is the angle between two intersecting curves? 

60. When several curves are to be joined, show what must be done to 

make them smooth curves. 

61. Define intersection of lines. 

62. Show that the tangent intersects the curve at a zero angle. 

63. Define order of contact of tangents. 

64. If three points become coincident on tangcncj^ what order of contact 

does the tangent have? 

65. Define osculating circle. 

66. Define centre of curvature. 

67. Is the radius of the osculating circle to a curve the radius of curvature 

at that point? 
08. Show how an osculating circle maj' have second order contact with 
a plane curve. 

69. What is an osculating plane? 

70. When is the osculating phuie the i^lane of the curve? 

71. Define point of inflexion. 

72. What is an infiexional tangent? 

73. Does the radius of curvMture change from one side of the curve to 

the other at a point of inflexion? 

74. Define princii)al normal. 

75. Show that the centr(\s of curNalmc^ a( a point of inflexion lie on 

opposite sides of the normal to the cuixc liirough the point of 
inflexion. 



164 GEOMETRICAL PROBLEMS IX PROJECTION 

76. When a normal is drawn to a curve is the one in the osculating plane 

the one generally understood? 

77. How many normals may be dra^^^l to a doubly cun-ed Une at a 

given point? 

78. What is meant by rectification? 

79. Rectify a 2" diameter circle. Compute the length of the rectified 

circle (=2x3.1416) and express the ratio of rectified to computed 
length as a percentage. 

80. Define involute and evolute. 

81. Show that all involutes to a cur\-e are parallel curv'es. 

82. Show that the involute is always normal to the evolute at the point 

for wliich it corresponds. 

83. Show that the dra^^ing of the evolute is the reverse process of drawing 

the involute. 

84. Draw the involute of a circle. 

85. Draw the involute to an elUpse. • 

86. Draw the evolute to an elhpse. 

87. Draw the involute to a parabola. 

88. Draw the evolute to a parabola. 

89. Draw the involute to one branch of a hyperbola. 

90. Draw the evolute to one branch of a hyperbola. 

91. Draw the involute to a cycloid. 

92. Draw the evolute to a cycloid. 

93. Draw the involute to an epicycloid. 

94. Draw the evolute to an epicycloid. 

95. Draw the involute to a hA-pocycloid. 

96. Draw the evolute to a hypocycloid. 

97. Draw the involute to an Archimedian spiral. 

98. Draw the evolut-e to an Archimedian spiral. 



CHAPTER X 

CLASSIFICATION OF SURFACES 

1001. Introductory. A surface may be generated by the 
successive positions of a line which moves so as to generate an 
area. As there are infinite varieties of hnes and as their motion 
may again be in an infinite variety of ways, therefore, an infinite 
variety of possible surfaces result. In engineering it is usual 
to limit the choice of surfaces to such as may be easily reproduced 
and easily represented. Surfaces, like lines and points, are 
mathematical concepts because they have no material exist- 
ence. 

When curved surfaces, of a more or less complex nature, are 
to be represented, they may be shown to advantage, by the 
effects of light on them. Examples of this kind are treated in 
Chapters XIV and XV. 

1002. Plane Surface. A plane surface is a surface such 
that when any two points in it are joined by a straight line, the 
line lies wholly within the surface. Thus, three points may be 
selected in a plane and two intersecting lines may be drawn 
through the three points; the intersecting lines lie in the plane 
and, therefore, may be used to determine it. Also, a line and 
an external point may determine a plane. 

The plane surface may also be conceived as being generated 
by a straight line, moving so as to touch another lino, and con- 
tinually remaining parallel to its original position. Hence, 
also, two parallel lines determine a plane. 

In the latter case, the moving straight lino may bo consid- 
ered as a rectilinear generatrix, touching a roctilinoar directrix, 
and occupying, consecutive positions in its motion. Any one 
position of the generatrix may be used as an element of the 
surface. 

165 



166 



GEOMETRICAL PROBLEMS IN PROJECTION 



Fig. 164 shows a plane surface ABCD on which straight Unes 
ab, cd, ef and gh are drawn, all of which must lie wholly within 
the plane, irrespective of the direction in which they are drawn. 
Any curve drawn on this surface is a plane curve. 

1003. Conical surface. If a straight line passes through 
a given point in space and moves so as to touch a given fixed 
curve, the surface so generated is a conical surface. The straight 
line is the rectilinear generatrix, the fixed point is the vertex 
and the given fixed curve is the directrix, which need not be a 
closed curve. The generatrix in any one position is an element 
of the surface. 

Fig. 165 shows a conical surface, generated in the manner 





■ 



Fig. 164. 



Fig. 165. 



indicated. Either the upper or the lower curve may be consid- 
ered as the directrix. In fact, any number of fines may be drawn 
on the resulting surface, whether the lines be singly or doubly 
curved, and any of which will fill the office of directrix. A plane 
curve is generally used as the directrix. 

With a generatrix of indefinite extent, the conical sur- 
face generated is a single surface (not two surfaces as might 
appear) ; the vertex o is a point of union and not of separa- 
tion. The portion of the surface from the vertex to either 
side is called a sheet; hence, there are two sheets to a 
conical surface. 

1004. Cone. The cone is a sohd, bounded by a closed conical 
surface of one sheet and a plane cutting all the elements. The 



CLASSIFICATION OF SURFACES 



167 



curve of intersection of the rectilinear elements and the plane 
cutting all the elements is the base. A circular cone has a circle 
for its base and the line joining the vertex with the centre of the 
base is the axis of the cone. If the axis is perpendicular to the 
plane of the base the cone is a right cone. When the base is a 
circle, and the axis is perpendicular to the plane of the base, 
the cone is a right circular cone or a cone of revolution.* A 
cone of revolution may be generated by revolving a right tri- 
angle about one of its legs as an axis. The hypothenuse is then 
the slant height of the cone. The perpendicular distance from 
the vertex to the plane of the base is the altitude of the cone. 
The foot of the perpendicular may fall outside of the centre of 
the base and in such a case, the cone is an oblique cone. 

The frustum (plural : — frusta) of a cone is the limited portion 
of the solid bounded by a closed conical surface and two parallel 
planes, each cutting all the elements, and giving rise to the upper 
base and the lower base of the frustum of a cone. The terms 
upper and lower base are relative; it is usual to consider the 
larger as the lower base and to represent the figure as resting 
on it. When the cutting planes are not parallel, then the solid 
is a truncated cone. 

1005. Representation of the cone. 

other object, is represented by its projec- 
tions on the principal planes. For con- 
venience in illustrating a cone, the plane 
of the base is assumed perpendicular to 
one of the principal planes, as then its 
projection on that plane is a line. Fig. 
166 shows a cone in orthographic projec- 
tion. The vertex O is shown by its 
projections o and o'; d'c' is the vertical 
projection of the ])ase since the plane of 
the base is assumed perpendicular to the 
vertical plane. The extreme limiting ele- 
ments o'c' and o'd' are also sliown, thus 

completing the vertical projection. In the horizontal projiH'tion, 
any curve acbd is assumed as the projection of the base so 

* This (li.slin(!ti()n is nuulo hocaus(» ti coiu* with an clliplical has(» may 
also bo a i'\)i}\i cono vvIumi tlio vert ox is chosen so that it is on a por|UM\(liciihir 
to the phiiH' of the base, at the intcrsiM't ion of (he inajtn" and minor axo.^. 



A cone, like any 




Fu;. 100. 



168 



GEOMETRICAL PROBLEMS IN PROJECTION 



that d and c are corresponding projections of d' and c'. From 
o, the Hnes ob and oa are drawn, tangent to acbd, thus completing 
the horizontal projection. 

It must here be emphasized, that acbd is not the actual base 
of the cone, but only its projection. It is impossible to assume 
two curves, one in each plane of projection, and call them corre- 
sponding projections of the same base. The corresponding points 
must be selected so that they will lie in one plane, and that 
plane must be the plane of the base. If it be desired to show 
the base in both projections, when the plane of the base is in- 
clined to the principal planes, it is necessary to assume one pro- 
jection of the base. Lines are then drawn in that plane, through 
the projection of the base and the corresponding projections 






Fig. 167. 



Fig. 168. 



of the lines are found. The points can then be determined as 
they must be situated on these lines. (Arts. 704 and 811.) 

1006. To assume an element on the surface of a cone. 

To assume an element of a cone, assume the horizontal projec- 
tion oa in Fig. 167. There are two elements on the cone which 
have the same horizontal projection and they are shown as 
o'a' and o'b' in the vertical projection. If oa be considered 
visible, while viewing the horizontal plane, then o'a' is its corre- 
sponding projection. If o'b' be the one assumed projection 
then oa is on the far side and should in this case be drawn dotted. 

1007. To assume a point on the surface of a cone. To 

assume a point on the surface of a cone, assume c' in Fig. 168 
as the vertical projection, somewhere within the projected area. 



I 



CLASSIFICATION OF SURFACES 



169 



Draw the element o'b' through c' and find the corresponding 
projection of the element. If o'b' is visible to the observer, 
then ob is the corresponding projection, and c on it is the required 
projection. If o'b' is on the far side, then d is the desired pro- 
jection. 

A slightly different case is shown in Fig. 169. If c is assumed 
on the visible element oa then c' is the corresponding projection 
on o'a'. Otherwise, if ob is dotted (invisible) then o'b' is the 
corresponding element and d and d' are corresponding projections. 

1008. Cylindrical surface. When a straight line moves 
so that it remains continually parallel to itself and touches a 
given fixed curve, the surface generated is a cylindrical surface. 





Fig. 169. 



Fig. 170. 



The straight line is the rectilinear generatrix, the fixed curve 
is the directrix and need not be a closed curve. The generatrix 
in any one position is an element of the surface. 

Fig. 170 shows a cylindrical surface, generated in the manner 
indicated. Any curve, drawn on the resultant surface, whether 
singly or doubly curved, may be considered as the directrix. 
The limiting curves that are shown in the figure may also be used 
as directrices. A plane curve is generally used as a directrix 

1009. Cylinder. A cylinder is a solid bounded by a closed 
cylindrical surface and two parallc^l pianos cutting all the elonunits. 
The planes cut curves from cylindrical surface wliich form the 
bases of the cylinder and may be termed upper and lower bases 
if the cylinder is so situated that the nonu^nclatun^ fits. WIumi 
the planes of the bases are not parallel t hen it is called a truncated 
cylinder. 



170 



GEOMETRICAL PROBLEMS IX PROJECTION 



Should the bases have a centre, a figure such as a circle for 
instance, then a line joining these centres is the axis of the cylinder. 
The axis must be parallel to the elements of the cylinder. If the 
axis is inclined to the base, the cylinder is an oblique cylinder. 
On the other hand, if the axis is perpendicular to the plane of 
the base, it is a right cylinder and when the base is a circle, it is 
a right circular cylinder, or, a cylinder of revolution. The 
cylinder of revolution ma} be generated by revolving a rectangle 
about one of its sides as an axis. A right cylinder need not have 
a circular base, but the elements must be perpendicular to the 
plane of the base. 

1010. Representation of the cylinder. A cylinder, 
represented orthographically is sho^\TL in Fig. 171. Suppose 
the base is assumed in the horizontal plane, then e'g' may be 

taken as the vertical projection of the base. 
Also e'f and g'h', parallel to each other, 
may be taken as the projections of the 
extreme Imiiting elements. Any curve, as 
aecg. may be dra^^m for the horizontal 
projection so long as e and g are corre- 
sponding projections of e' and g'. In addi- 
tion, draw ab and cd parallel to each other 
and tangent to the curve aecg. The hori- 
zontal projection is thus completed. It is 
to be noted that aecg is the true base 
YiQ, 171. because it hes in the horizontal plane. If 

the plane of the base does not coincide 
TN-ith the horizontal plane, then, as in Art. 1005. what appHes 
to the selection of the projection of the base of the cone apphes 
here. 

1011. To assume an element on the surface of a cylinder. 

To assume an element on the surface of a cyhnder, select any 
line, ab. Fig. 172, as the horizontal projection. As all parallel 
lines have parallel projections, then ab must be parahel to the 
extreme elements of the cylinder. If ab is assumed as a \'isible 
element, then a'b' is its corresponding projection, and is shown 
dotted, because hidden from view on the vertical projection. 
If c'd' is a \asible element, then cd should be dotted in the 
horizontal projection. 




CLASSIFICATION OF SURFACES 



171 



1012. To assume a point on the surface of a cylinder. 

To assume a point on the surface of a cylinder, select any point 
c, Fig. 173, in the horizontal projection, and draw the element 
ab through it. Find the corresponding projection a'b' and on 
it, locate c', the required projection. What has been said before 
(Art. 1005) about the two possible cases of an assumed projection, 
applies equally well here and should require no further mention. 

1013. Convolute surface. A convolute surface is a surface 
generated by a line which moves so as to be continually tangent * 
to a line of double curvature. For purposes of illustration, 
the uniform cylindrical helix will be assumed as the line of double 




FiG. 172. 



Fig. 173. 



Fig. 174. 



curvature, remembering, in all cases, that the helix may be 
variable in radius and in motion along the axis, so that its char- 
acteristics may be imparted to the resulting convolute. 

The manner in which this surface is generated may be gained 
from what follows. Let abed. Fig. 174, be the horizontal pro- 
jection of a half portion octagonal prism on which is a piece of 
paper in the form of a right triangle is wound. The base of the 
triangle is therefore the perimeter of the prism and thehyjwth- 
enuse will appear as a broken line on the sides of the i)rism. 
If the triangle ])e unwound from the prism and the starting point 
of the curve described ))y the liypotlienuse on the horizontal 

* It may be obsorvod that tho tan^^ont to a liiu^ of (loubli« i-urvaturr must 
lie in the oKculatin^j; plane (Art. 925). 



172 



GEOMETRICAL PROBLEMS IN PROJECTION 



plane be at o, then the portion of the triangle whose base is oa 
will revolve about the edge a so as to describe the arc ol. At 
the point 1 the triangle is free along the face ab and now swings 
about b as a centre and describes the arc 
1-2. As the process goes on to the point 
2, the triangle is free on the face be and 
then swings about c as a centre and 
describes the arc 2-3; and so on. In the 
vertical projection, the successive positions 
of the hypothenuse are shown by a'l', 
b'2', c'3' etc. It will be noted that a'l' 
intersects b'2' at b', and b'2' intersects 
c'3' at c', etc.; but, a'l' does not intersect 
c'3' nor does b'2' intersect d'4'. Hence, the 
elements of the surface generated by the 
hypothenuse intersect two and two. The 
first element intersects the second; the 
second the third; the third the fourth, etc.; 
but the first does not intersect the third, or any beyond, nor does 
the second intersect the fourth or any elements beyond the fourth. 
When the prism approaches a cylinder as a limit by increas- 
ing the number of sides indefinitely, the hypothenuse wound 




Fig.- 174. 




Fig. i: 



Fig. 176. 



around the cyhnder approaches a hehx as a limit; the unwind- 
ing hypothenuse will become the generatrix, tangent to the helix, 
and will approach the desired convolute surface. The ultimate 
operation is a continuous one and may be seen in Fig. 175. The 
curve abed described by the h37)othenuse fd on the plane MM 



CLASSIFICATION OF SURFACES 



173 




Fig. 177. 



is the involute of a circle, if the cylinder is a right circular 
cylinder. 

Fig. 176 may indicate the nature of the surface more clearly. 
Examples of this surface may be 
obtained in the machine shop on 
observing the spring-like chips, 
that issue on taking a heavy cut 
from steel or brass. The surfaces 
are perhaps not exact convolutes, 
but they resemble them enough to 
give the idea. 

It is not necessary to have the 
tangent stop abruptly at the 
helix, the tangent may be a line 
of indefinite extent,, and, hence, 
the convolute surface extends both 
sides of the helical directrix. No 
portion of this surface intersects 
any other portion of the surface, 
but all the convolutions are dis- 
tinct from each other. Fig. 177 will perhaps convey the final 
idea. 

1014. Oblique helicoidal screw surface.* When the helical 
directrix of a convolute surface decreases in diameter, it will 

ultimately coincide ^vith the axis 
and tlie helix will become a line. 
Th{^ oblique helicoidal screw sur- 
face, therefore, resolves itself into 
the surface generated by a recti- 
linear generatrix revolving about 
another line which it intersects, at a constant angle, the inter- 
section moving along the axis at a uniform rate. The application 
of this surface is shown in the construction of the V thread 
screw which in order to become the United Statt^s Standard 
screw, must make an angle of ()()° at the V as shown in Fig. 17S. 

* Tho holicoid proper is a warpod surface (lOlG). If a straij2;hl lino 
touches two concentric helices of difl'ereiit (hameters and hes in a plane tangent 
to the inner hehx's cyhnder, the line will f2;enerate a warped surface. When 
the diameter of the cylinder l)(«coines '/.cm, the helix becomes a line and the 
helicoidal surfac^e is the same as that here ^;iven. 




Fig. 178. 



174 



GEOMETRICAL PROBLEMS IN PROJECTION 




Fig. 179. 



1015. Right helicoidal screw surface. If the diameter 
of the hehcal directrix still remains zero, and the rectilinear 

generatrix becomes perpendicular to 
the axis and revolves so that the in- 
tersection of the axis and the genera- 
trix move? along the axis at a constant 
rate.* a special case of the convolute is 
obtained. This special case of the 
convolute is called a right helicoidal 
screw surface, and when applied gives 
the surface of a square threaded screw as shown in Fig. 179. 

In both cases of the obhc^ue and right hehcoidal screw sur- 
faces the hehces at the outside and root (bottom) of the thread, 
are formed by the intersection of the screw surface and the 
outer and inner concentric cylinders. The pitch of both must 
be the same, as every point on the generatrix advances at a 
uniform rate. Hence, the angle of the tangent to the hehx 
must vary on the inner and outer cylinders. For this reason, 
the hehces have a different shape notwithstanding their equal 
pitch. 

1016. Warped surface. A warped surface is a curved 
siuiace, generated by a rectilineal' generatrix, mo^^.ng so that 
no two successive elements he in the same plane. Thus, the 
consecutive elements can neither be pai'allel nor intersect, hence, 
they are skew lines. An example of this smlace may be obtained 
by taking a series of straight sticks and drilling a smaU hole through 
each end of every stick. If a string be pa.ssed through each end 
and secured so as to keep them together, the series of sticks 
may be laid on a flat surface and thus represent successive 
elements of a plane. It may also be cur^^ed so as to represent 
a cylinder. Lastly, it may be given a twist so that no single 
plane can be passed through the axis of successive sticks: this 
latter case would then represent a warped surface. 

Warped surfaces find comparatively httle apphcation in 
engineering because they are difficult to construct or to dupUcate. 
At times, however, they are met with in the construction of 

* If the pitch becomes zero when the diameter of the helix becomes zero, 
it is the case of a line revolving about another Line, through a fixed point; 
the surface is therefore a cone of revolution, if the generatrix is inclined 
to the axis. If the generatrix is normal to the axis, the surface is a plane. 



CLASSIFICATION OF SURFACES 175 

" forms " for reinforced concrete work, where changes of shape 
occur as in tunnels, and similar constructions; in propeller screws 
for ships; in locomotive " cow-catchers," etc. 

1017. Tangent plane. If any plane be passed through the 
vertex of a cone, it may cut the surface in two rectilinear elements 
under which condition it is a secant plane. If this secant plane 
be revolved about one of the rectilinear elements as an axis, 
the elements of secancy can be made to approach so as to coincide 
ultimately. This, then, is a limiting position of the secant plane, 
in which case it becomes a tangent plane, having contact with 
the cone all along one element. 

If through some point on the element of contact, two inter- 
secting curved lines be drawn on the surface of the cone, then, 
also, two secants may be drawn to these curved lines and inter- 
secting each other at the intersection of the curved lines. The 
limiting positions of these secants will be tangent lines to the 
cone, and as these tangents intersect, they determine a tangent 
plane. The tangent plane thus determined is identical with 
that obtained from the limiting position of a secant plane. 

Instead of drawing two intersecting curves on the surface 
of the cone, it is possible to select the element of contact and any 
curve on the cone intersecting it. The tangent plane in this 
case is determined by the element of contact and the limiting 
position of one secant to the curve through the intersection of 
the element and the curve. 

As another example, take a spherical surface and on it draw 
two intersecting lines (necessarily curved). Through the point 
of intersection, draw two secants, one to each curve and deter- 
mine the tangent positions. Again, the plane of the two inter- 
secting tangents is the tangent to the sphere. Hence, as a general 
definition, a tang(Mit plane is the plane established by the limiting 
position of two intersecting secants as the ])oints of secancy 
reach coincidence. 

1018. Normal plane. Thc^ normal plane is any i)lane that 
is ])erpen(Hcuhir to tlie tangent ])lane. If a normal plane is to 
be drawn to a sphere at a given point, for instance, then construct 
the tangent i)lane and draw through tli(^ point of tangency any 
plane ixTpendicuhir to tlie tangcMit plane An infinite nunilier 
of noi-nial planes may be drawn, ail i)assing tlii'ongh the given 



176 GEOMETRICAL PROBLEMS IN PROJECTION 

point. The various normal planes will intersect in a common 
line, which is normal to the tangent plane at the point of tangency. 

1019. Singly curved surface. A singly curved surface is 
a surface whose successive rectilinear elements may be made 
to coincide with a plane. Hence, a tangent plane must be in 
contact all along some one rectilinear element. As examples, 
the conical, cylindrical, convolute and the helicoidal screw 
surfaces may be mentioned. 

1020. Doubly curved surface. A doubly curved surface is 
a surface whose tangent plane touches its surface at a point. 
Evidently, any surface which is not plane or singly curved must 
be doubly curved. The sphere is a familiar example of a doubly 
curved surface. 

1021. Singly curved surface of revolution. A singly^curved 
surface of revolution is a surface generated by a straight line 
revolving about another straight line in its own plane as an axis, 
so that every point on the revolving line describes a circle whose 
plane is perpendicular to the axis, and whose centre is in the axis. 
Thus, only two cases of singly curved surfaces can obtain, the 
conical and the cylindrical surfaces of revolution. 

1022. Doubly curved surface of revolution. A doubly 
curved surface of revolution is a surface generated by a plane 
curve revolving about a straight line in its own plane as an axis 
so that every point on the revolving curve describes a circle whose 
plane is perpendicular to the axis, and whose centre hes in the 
axis. Hence, there are infinite varieties of doubly curved sur- 
faces of revolution as the spherCj ellipsoid, hyperboloid, etc., 
generated by revolving the circle, ellipse, hyperbola, etc., about 
their axes. In the case of the parabola, the curve may revolve 
about the axis or the directrix in which cases two distinct types 
of surfaces are generated. Similarly with the hyperbola, the 
curve may generate the hyperboloid of one or two nappes depend- 
ing upon whether the conjugate or transverse axis is the axis of 
revolution, respectively. 

Sometimes, a distinction is made between the outside and 
inside surfaces of a doubly curved surface. For example, the 
outside surface of a sphere is called a doubly convex surface, 
whereas, the inside is an illustration of a doubly concave surface. 



CLASSIFICATION OF SURFACES 



177 



A circular ring raade of round wire, and known as a torus, is an 
example of a doubly concavo-convex surface. 

1023. Revolution of a skew line. An interesting surface 
is the one generated by a pair of skew 
lines when one is made to revolve about 
the other as an axis. Fig. 180 gives 
such a case, and as no plane can be 
passed through successive elements, it 
is a warped surface. While revolving 
about the axis, the line generates the 
same type of surface as would be gen- 
erated by a hyperbola when revolved 
about its conjugate axis. The surface 
is known as the hyperboloid of revo- 
lution of one nappe, and, incidentally, 
is the only warped surface of revolu- 
tion. 

1024. Meridian plane and me- 
ridian line. If a plane be passed 
through the axis of a doubly curved 
surface of revolution, it will cut from 
the surface a line which is the me- 
ridian line. The plane cutting the 
meridian Une is called the meridian 

plane. Any meridian line can be used as the generatrix of the sur- 
face of revolution, because all meridian lines are the same. The 
circle is the meridian line of a sphere, and for this particular 
surface, every section is a circle. In general, 
every plane perpendicular to the axis will cut a 
circle from any surface of revolution, whetlier 
singly or doubly curved. 

1025. Surfaces of revolution having a 
common axis. If two surfaces of revolution 
have a common axis and the surfaces intersect, 
tangentially or angularly, they do so all arountl 
in a circle which is common to the two surfaces 
of revolution. Thus, the two surfaces sliown in 
Fi<»;. ISl intersect, angularly, in a circle having ab as a diameter, 
and intersect, tangentially, in a circle having cd as a dianuMer. 




Fig. 180. 




Fic. LSI. 



178 



GEOMETRICAL PROBLEMS IN PROJECTION 



1026. Representation of the doubly curved surface 
of revolution. Fig. 182 shows a doubly curved surface of 
revolution shown in two views. One view shows the same as 
that produced by a meridian plane cutting a meridian line and 
the other shows the same as concentric circles. When considered 
as a sohd, no ground line is necessary as the distance from the 
principal planes is unimportant. Centre lines, ab, cd and ef 




Fig. 182. 

should be shown, ab and ef being represented as two lines, be- 
cause both views are distinct from each other. The lines indi- 
cate that the object is symmetrical about the centre line as an 
axis. 




1027. To assume a point on a doubly curved surface of 
revolution. Let c, Fig. 183, be assumed as one point on the 

surface.* With oc as a radius, 
draw the arc ca, a is, there- 
fore, the revolved position of 
c when the meridian plane 
through CO has been revolved 
to ao. Hence, a is at a' or 
a" in the corresponding view. 
On counter-revolution, a' de- 
scribes a circle, the plane of which is perpendicular to the axis, 
and the plane is shown by its trace a'c' ; in the other view, a 
returns to c and, hence, c' is the final position. It may also be 
at Q," for the same reason, but then c is hidden in that view. If, 
on the other hand, d is chosen as one projection, its corre- 

* These views bear third angle relation to each other. 



Fig. 183. 



CLASSIFICATION OF SURFACES 179 

spending projection is d', if d is visible; or, it is d", if d is 
hidden. 

1028. Developable surface. When a curved surface can 
be rolled over a plane surface so that successive elements come 
in contact with the plane and that the area of the curved surface 
can be made to equal the plane surface by rectification, the 
surface is a developable one. Hence, any singly curved surface, 
like a cylinder, can be rolled out flat or developed. A sphere 
cannot be rolled out as a flat surface because it has point con- 
tact with a plane, and is, therefore, incapable of development. 
If a sphere is to be constructed from flat sheets, it may be ap- 
proximated by cutting it into the type of slices called lunes, 
resembling very much the slices made by passing meridian planes 
through the axis of a sphere. To approach more nearly the 
sphere, it would be necessary to take these lunes and hammer 
them so as to stretch the material to the proper curvature. Simi- 
larly, in the making of stove-pipe elbows, the elbows are made 
of limited portions of cylinders and cut to a wedge shape so as 
to approximate the doubly curved surface known as the torus. 

1029. Ruled surface. Every surface on which a straight 
line may be drawn is called a ruled smiace. A ruled surface 
may be plane, singly curved or doubly curved. Among the singly 
curved examples may be found the conical, cylindrical, convolute 
and helicoidal screw surfaces. The hyperboloid of revolution 
of one nappe furnishes a case of a doubly curved ruled surface 
(1023). 

1030. Asymptotic surface. If a hyperbola and its as>Tnp- 
totes move so that their plane continually remains parallel to 
itself, and any point on the curve or on the asymptotes touches 
a straight line as a directrix, the hyperbola will generate a hy- 
perbolic cylindrical surface and the asymptotes will generate 
a pair of asymptotic planes. Also, if the hyperbola revolves 
about the transverse or conjugate axis, the hyperbola will generate 
a hyperboloid of revolution and the asymptote will generate a 
conical surface which is asymptotic to the hyperboloid. In all 
cases, the asymptotic surface is tangent at two lines, straight or 
curved, at an inlinit(' distance apart and the surface passes within 
finite distance of the intersection of the axes of the curve. 



180 



GEOMETRICAL PROBLEMS IN PROJECTION 



1031. Classification of surfaces. 



Surfaces. ■ 



Ruled surfaces. 

Straight lines 
may be drawn on 
the resulting sur- 
face. 



Doubly curved 
surfaces. Tan- 
gent plane 
touches surface 
at a point. 



Singly curved sur- 
faces. May be 
developed into a 
flat surface by rec- 
tification. 



Planes. Any two 

points when joined 

by a straight line he 

whoUy within the 

surface. 

r C n i c a 1 surfaces. 
Rectilinear ele- 
ments pass through 
a given point in 
space and touch a 
curved directrix. 

Cylindrical surfaces. 

Rectilinear ele- 

ments are parallel 
to each other and 
touch a curved di- 
rectrix. 

Convolute surfaces. 

Rectilinear ele- 
ments tangent to a 
line of double curva- 
ture. Consecutive 
elements intersect 
two and two; no 
three intersect in a 
common point. 
Warped surfaces. 
No two consecutive 
elements lie in the 
same plane; hence, 
they are non-devel- 
opable. 

Doubly curved surfaces of revolution. 
Generated by plane curves revolving about 
an axis in the plane of the curve. All 
meridian lines equal and all sections per- 
pendicular to axis are circles. 

Unclassified doubly curved surfaces. All 
others which do not fall within the fore- 
going classification. 



QUESTIONS ON CHAPTER X 



1. How are surfaces generated? 

2. What is a plane surface? 

3. What is a rectihnear generatrix? 

4. What is a directrix? 

5. What is an element of a surface? 

6. What is a conical surface? 



CLASSIFICATION OF SURFACES 181 

7. What is the directrix of a conical surface? 

8. What is the vertex of a conical surface? 

9. Is it necessary for the directrix of a conical surface to be closed? 

10. What is a nappe of a cone? How many nappes are generated in a 

conical surface? 

11. What is a cone? 

12. What is the base of a cone? 

13. Must all elements be cut by the plane of the base for a cone? 

14. What is a circular cone? 

15. What is the axis of a cone? 

16. What is a right circular cone? Is it a cone of revolution? 

17. What is the altitude of a cone? 

18. What is the slant height of a cone? 

19. What is the an oblique cone? 

20. What is a frustum of a cone? 

21. How are the two bases of a frustum of a cone usually designated? 

22. What is a truncated cone? 

23. In the representation of a cone, why is the plane of the base usually 

assumed perpendicular to the plane of projection? 

24. Is it necessary that the base of a cone should be circular? 

25. Draw a cone in orthograpliic projection and assume the plane of 

the base perpendicular to the vertical plane. 

26. Draw a cone in projection and show how an element of the surface 

is assumed in both projections. State exactty where the element 
is chosen. 

27. Draw a cone in projection and show how a point is assumed on its 

surface. Locate point in both projections. 

28. What is a cylindrical surface? 

29. Define generatrix of a cylindrical surface; directrix; element. 

30. Is it necessary that the directrix of a cylindrical surface be a closed 

curve? 

31. How is a cylinder differentiated from a cylindrical surface? 

32. How many bases must a cyhnder have? 

33. What is the aKis of a cylinder? 

34. Must the axis of the cylinder be i)arallel to the elements? Wliy? 

35. What is an oblicjue cyhnder? 

36. What is a right circular cylinder? Is this cylinder a cylinder of 

revolution? 

37. Is it necessary that a right cylindcT have a circle for the base? Wii}'? 

38. Draw an oblique cylinder whose base hes in the horizontal plane. 

39. In Question 38, assume an element of the surface antl state whii'h 

element is chosen. 

40. Which elements are the limiting (>l(Miu>nts in (^luvstion 38? 

41. Draw a cylinder in projection and tluMi assume a point on the surfaci* 

of it. Show it in both i)rojections. 

42. What is a convolute surface? 

43. What is an ol)li(iue helicoidal screw surface? Uive a i)roniinent 

example of it. 



182 GEOMETRICAL PROBLEMS IN PROJECTION 

44. What is a right hehcoidal screw surface? Give a prominent example 

of it. 

45. Show that the hehcoidal screw surfaces are limiting helicoids as the 

inner hehx becomes of zero diameter. 

46. What is a warped surface? Illustrate by sticks. 

47. What is a secant plane? 

48. Show how a tangent plane is the hmiting position of a secant plane 

to a conical surface. 

49. Show how a tangent plane is the hmiting position of a secant plane 

to a cylindrical surface. 

50. Show that a tangent hne to the surface is the limiting position of a 

secant drawn to a curve on the surface. 

51. Show how two intersecting hues may be drawn on a curved surface 

and how the Hmiting positions of two secants drawn through 
this point of intersection determine a tangent plane to the 
surface. 

52. Show how one element of a conical surface and one hmiting posi- 

tion of a secant determine the tangent plane to the conical 
surface. 

53. Show how one element of a cyUndrical surface and one hmiting 

position of a secant determine the tangent plane to the cyhndrical 
surface. 

54. Show how two intersecting Hues may be draT\'n on the surface of a 

sphere and how the hmiting positions of two secants dra^Ti through 
the intersection determine a tangent plane to the sphere. 

55. Define tangent plane in terms of the two intersecting tangent lines 

at a point on a surface. 

56. Define normal plane. 

57. How many normal planes may be dra\Mi through a given point 

on a surface? 

58. What is a normal (hne) to a surface? 

59. Define singly curved surface. Give examples. 

60. Define doubl}^ curved surface. Give examples. 

61. Define singlj^ curved surface of revolution. Give examples. 

62. Define doubly curved surface of revolution. Give examples. 

63. What is a doubl}^ convex surface? Give examples. 

64. What is a doubty concave surface? Give examples. 

65. What is a doubly concavo-convex surface. Give examples. 

66. Describe the surface of a torus. Is this a concavo-convex surface? 

67. What surface is obtained when a pair of skew hnes are revolved 

about one of them as an axis? 

68. Construct the surface of Question 67. 

69. What is a meridian plane? 

70. What is a meridian hne? 

71. Wh}'' can any meridian hne be assumed as a generatrix for its partic- 

ular surface of revolution? 

72. What curves are obtained by passing planes perpendicular to the 

axis of revolution? 



CLASSIFICATION OF SURFACES 183 

73. Wlien two surfaces of revolution have the same axis, show that the 

intersection is a circle whether the surfaces intersect tangentially 
or angularly. 

74. Show how a doublj^ curved surface of revolution may be represented 

without the ground line. Draw the proper centre hnes. 

75. Assume a point on the surface of a doubly curved surface of revolu- 

tion. 

76. What is a developable surface? 

77. Is development the rectification of a surface? 

78. Are singly curved surfaces developable? 

79. Are doubly curved surfaces developable? 

80. Are warped surfaces developable? 

81. What is a ruled surface? Give examples. 

82. What is an asymptotic plane? Give an example. 

83. If a hyperbola and its asymptotes revolve about the transverse 

axis, show why the asymptotic lines generate asymptotic cones 
to the resulting hyperboloids. 

84. Show what changes occur in Question 83 when the conjugate axis 

is used. 

85. Make a classification of surfaces. 



CHAPTER XI 



intersections of sutfaces by plaxe.^ 
de\t:lop-MExt 



AND THEIR 



1101. Introductory. When a line is inclined to a plane, 
it will if sufficiently produced, pierce the plane in a point. 
The general method involved has been shown (Art. 823 i 
for straight lines, and consists of passing an auxiliary' plane 
through the given line, so that it cuts the given plane in a line 
of intersection. The piercing point must be somewhere on this 
line of intersection and also on the given line ; hence it is at their 
intersection. 

In the case of doubly curved lines, the passing of auxiliarj^ 
planes through them is evidently impossible. Cur^'ed surfaces, 
instead of planes, are therefore used as the auxiliary' surfaces. 
Let, for example. Fig. 1S4 show a doubly curved line, and let the 

object be to find the piercing 
point of the doubly cur^'ed line 
on the principal planes. If a 
cylindrical surface be passed 
through the given line, the 
elements of which are perpen- 
dicular to the horizontal plane, 
it v.ill have the curve ab as 
its trace, which will also be the 
horizontal projection of the 
cur\'e AB in space. Similarly, the vertical projecting cylindrical 
surface will cut the vertical plane in the line a'b', and will be the 
vertical projection of the cur^^e AB in space. If a perpendicular be 
erected at a, where ab crosses the ground line, it will intersect the 
vertical projection of the curve at a', the vertical piercing point of 
the curve. The entire process in substance consists of this: the 
surface of the horizontal projecting cylinder cuts the vertical 
plane in the line aa'; the piercing point of the cur^'e AB must 

1S4 




Fig. 1S4. 



INTERSECTIONS OF SURFACES BY PLANES 185 

lie on aa' and also on AB, hence it is at their intersection a'. 
Likewise, b, the horizontal piercing point is found by a process 
identical with that immediately preceding. 

1102. Lines of intersection of solids by planes. The 

extension of the foregoing is the entire scheme of finding the 
line of intersection of any surface with the cutting plane. Ele- 
ments of the surface pierce the cutting plane in points; the locus 
of the points so obtained, determine the line of intersection. 

A distinction must be made between a plane cutting a sur- 
face, and a plane cutting a solid. In the former case, the surface', 
alone, gives rise to the line of intersection, whether it be an open 
surface, or a closed surface; the cutting plane intersects the 
surface in a line which is the line of intersection. In the latter 
case, the area of the solid, exposed by the cutting plane, is a 
section of the solid. This is the scheme of using section planes 
for the elucidation of certain views in drawing (313). 

1103. Development of surfaces. The development of a 
surface consists of the rolling out or rectification of the surface 
on a plane, so that the area on the plane is equal to the area 
of the surface before development. If this flat surface be rolled 
up, it will regenerate the original surface from which it has been 
evolved (1028). 

For instance, if a flat rectangular sheet of paper be rolled 
in a circular form, it will produce the surface of a cylinder of 
revolution. Similarly, a sector of a circle may be wound up 
so as to make a right circular cone. In both cases, the flat sur- 
face is the development of the surface of the cylinder or cone 
as the case may be. 

1104. Developable surfaces. A prism may be rolled over 
a flat surface and each face successively comes in intimate con- 
tact with the flat surface; hence, its surface is developable. A 
cylinder may likewise be rolled over a flat surface ami \hv con- 
secutive elements will successively coincide with the flat sur- 
face, this, again, is therefore a devel()])abl(^ surface. The sur- 
faces of the pyramid and the cone are also (k^velopable for similar 
reasons. 

A s])her(% wh(^n rolh^l ov(M' a flat suifac(\ touches at only 
one i)oini and not along any one ("hMncnl ; hence, its surface it; 



186 GEOMETRICAL PROBLEMS IN PROJECTIOX 

not developable. In general, any surface of revolution which 
has a curvihnear* generatrix is non-developable. 

A warped surface is also a type of surface which is non-develop- 
able, because consecutive elements even though rectilinear, 
are so situated that no plane can contain them. 

Hence, to review^, only singly curved surfaces, and such 
surfaces as are made up of intersecting planes, are developable. 
Doubly curved surfaces of any kind are only developable in an 
approximate way, by dividing the actual surface into a series 
of developable surfaces; the larger the series, the nearer the 
approximation.! 

1105. Problem 1. To find the line of intersection of the 
surfaces of a right octagonal prism with a plane inclined to its 
axis. 

Construction. Let abed, etc., Fig. 185, be the plan of the 
prism, resting, for convenience, on the horizontal plane. The 
elevation is shown by a'b'c^ etc. The plane T is perpendicular 
to the vertical plane and makes an angle a with the horizontal 
plane. In the supplementary view S, only half of the intersec- 
tion is shown, because the other half is symmetrical about the 
line 2i"e"'. Draw from e'd'c'b' and a', lines perpendicular 
to Tt' and draw a'^'e'^'', anywhere, but always parallel to Tt'. 
The axis of symmetry ea, shown in plan, is the projection of 
e'a' in elevation, and e'a' is equal to e'^'a^''; the points e'" and 
2i" are therefore established on the axis. To find &"\ draw 
dm, perpendicular to ea; dm is shown in its true length as it is 
parallel to the horizontal plane. Accordingly, set off m'''d''' = 
md, on a line from d^ perpendicular to Tt', from e'^'a''' as a 
base line. As nb = md, then n'''b'''=m'''d''', and is set off 
from e'^'a''', on a line from b', perpendicular to Tt'. The final 
point c'" is located so that o'''c''' = oc and is set off from e'^'a''', 
on a line from c', perpendicular to Tt. The ^'half section" is 

* If a curvilinear generatrix moves so that its plane remains continually 
parallel to itself and touches a rectilinear directrix, the surface generated 
is a cylindrical surface. Hence, this surface cannot be included in this 
connection. 

t Maps are developments of the earth's surface, made in various ways. 
This branch of the subject falls under Spherical Projections. The student 
who desires to pursue this branch is referred to the treatises on Topographical 
Drawing and Surveying. 



INTERSECTIONS OF SURFACES BY PLANES 



187 



then shown completely and is sectioned as is customary. The 
line of intersection is shown by e'''(i'''c'''b'''a'" ; the half section 
is the area included by the lines e'''d"'c'"b"'a"Vo"We"'. 

1106. Problem 2. To find the developed surfaces in the 
preceding problem.* 

Construction. As the prism is a right prism, all the vertical 
edges are perpendicular to the base; the base will develop into a 
straight line and the vertical edges will be perpendicular to it, 




Fig. 185. 



spaced an equal distance apart, because all faces are equal. 
Hence, on the base line AA, Fig. 185, lay off the perimeter of the 
prism and divide into eight equal parts. The distance of a" 
above the base line AA is equal to the distance of a' al)ove 
the horizontal plane; b'' is above AA, a distance ecjual to b' 
above the horizontal plane. In this way, all })oints are located. 
It will be observed that the development is synnnetrical about 
the vertical through e". 

To prove the accuracy of the construction, the developed 
surfa(;e may be laid out on paper, creased along all tlu* verticals 

* In the problems to follow, the bases are not included in the development 

JIM Ihoy arc ovidcul from tho drawing!;. 



188 



GEOMETRICAL PROBLEMS IN PROJECTION 



and wound up in the form of the prism. A flat card will admit 
of being placed along the cut, proving that the section is a plane 
surface. 

The upper and lower portions are both developed; the dis- 
tance between them is arbitrary, only the cut on one must exactly 
match with the cut on the other. 

1107. Problem 3. To find the line of intersection of the 
surface of a right circular cylinder with a plane inclined to its 
axis. 

Construction. Let Fig. 186 show the cylinder in plan and 




elevation. The plane cutting it is an angle a with the horizontal 
plane and is perpendicular to the vertical plane. It is customary, 
in the application, to select the position of the plane and the 
object so that subsequent operations become most convenient, 
so long as the given conditions are satisfied. Pass a series of 
auxiliary planes through the axis as oe, od, etc., spaced, for con- 
venience, an equal angular distance apart. These planes cut 
rectiUnear elements from the cylinder, shown by the verticals 
through e'd'c', etc. Lay off an axis e'"a''^ parallel to Tt', so that 
€" e' and a'''a' are perpendicular to Tt'. To find 6!" , it is 



INTEKSECTIONS OF SURFACES BY PLANES 1S9 

known that md is shown in its true length in the plan; hence, lay 
off m'''d'''=m(i from the axis e"'a''', on a line from d', perpen- 
dicular to Tt'. Also, make o'''c'" = oc and n'''b''' = nb in the same 
way. Draw a smooth curve through ^'"^"q!"^"^!"^'"^!"^!"^'" 
which will be found to be an ellipse. The ellipse may be drawn 
by plotting points as shown, or, the major axis ^'"W" and the 
minor axis c"'c"' may be laid off and the ellipse drawn by any 
method.* Both methods should produce identical results. 
The area of the ellipse is the section of the cylinder made by 
an oblique plane and the ellipse is the curve of intersection. 
As an example, a cylindrical glass of water may be tilted to the 
given angle, and the boundary of the surface of the water will 
be elliptical. 

1108. Problem 4. To find the developed surface in the 
preceding problem. 

Construction. As the cylinder is a right cylinder, the ele- 
ments are perpendicular to the base, and the base will develop 
into a straight line, of a length, equal to the rectified length 
of the circular base. Divide the base AA Fig. 186, into the same 
number of parts as are cut by the auxiliary planes. Every 
element in the elevation is shown in its true size because it is 
perpendicular to the horizontal plane; hence, make a''A equal 
to the distance of a! above the horizontal plane; b''A equal 
to the distance of b' above the horizontal plane, etc. Draw a 
smooth curve through the points so found and the development 
will appear as shown at D in Fig. 186. Both upper and lower 
portions are shown developed, either one may be wound up like 
the original cylinder and a flat card placed 
across the intersection, showing that the sur- 
face is plane. 




1109. Application of cylindrical sur- 
faces. Fig. 187 shows an elbow, approximat- 
ing a torus, made of sheet metal, by the use a^\ 
of short sections of cylinders. The ellipse is 
symmetrical about ])oth axes, and, hence, the lH;. is7. 

upper portion of the cylinder may be added to 
the lower portion so as to give an offset. Indeed in this way olbowj: 

♦ See Art. 901). 



190 



GEOMETRICAL PROBLEMS IN PROJECTION 




Fig. 188. 



are made in practice. The torus is a doubly curved surface, 
and, hence, is not developable, except by the approximation 

shown. To develop these individual 
sheets, pass a plane ab, perpendicular 
to the axis. The circle cut thereby 
will then develop into a straight 
line. Add corresponding distances 
above and below the base line so 
established and the development 
may be completed. Fig. 188 shows 
the sheets as they appear in the 
development. 

A sufficient number of points 

must always be found on any curve, 

so that no doubt occurs as to its form. In the illustration, the 

number of points is always less than actually required so as to 

avoid the confusion incident to a large number of construction lines. 

1110. Problem 5. To find the line of intersection of the 
surfaces of a right octagonal pyramid with a plane inclined to 
its axis. 

Construction. Let Fig. 189 represent the pyramid in plan 
and elevation, and let T be the cutting plane. The plane T 
cuts the extreme edge o'e' at e', horizontally projected at e, 
one point of the required intersection. It also cuts the edge 
o'd' at d', horizontally projected at d, thus locating two points 
on the , required intersection. In the same way b and a are 
located. The point c cannot be found in just this way. If the 
pj'Tamid be turned a quarter of a revolution, the point c' will be 
at q' and c'q' \\dll be the distance from the axis where the edge 
o'c' pierces the plane T. Hence, lay off oc = c'q' and complete 
the horizontal projection of the intersection. 

To find the true shape of the intersection, draw the supple- 
mentary view S. Lay off e'^'a''' as the axis, parallel to Tf; 
the length of the axis is such that e'a' = e"V. From the 
horizontal projection, dm is made equal to &"'m!" in the supple- 
mentary view, the latter being set off from the axis e'"a'", and 
on a line from d', perpendicular to Tt'. Also, co = c"'o''' and 
bn = b"V. Thus, the true intersection is shoTvn in the supple- 
mentary view S. 



INTERSECTIONS OF SURFACES BY PLANES 



191 



1111. Problem 6. To find the developed surfaces in the 
preceding problem. 

Construction. The extreme element o's\ Fig. 189, is 
shown in its true length on the vertical plane because it is parallel 
to that plane. Accordingly, with any point o" as a centre, 
draw an indefinite arc through AA, so that o''A = o's'. Every 
edge of the pyramid meeting at the vertex has the same length 
and all are therefore equal to o''A. The base of each triangular 
face is shown in its true length in the plan, and, hence, with any 
one of them as a length, set the distance off as a chord on AA 




Fig. 189. 



by the aid of a divider, so that the number of steps is equal to 
the number of faces. Draw these chords, indicate the edges, 
and it then remains to show where the cutting plane intersects 
the edges. The extreme edge o'e' is shown in its true length, 
hence, lay off o''e"=o'e'. The edge o'd' is not shown in its 
true length, but if the pyramid is revolved so that this edge 
reaches the position o's', then d' will reach p' and o'p' is the 
desired true length as it is now parallel to the vortical plane; 
therefore, set off ©V = o"d". In this way, o'q' = o"c", and o'r' = 
o"b". The edge o'a' is shown in its true length, therefore, o'a' 
= o"a''; and a"A on one side must equal a"A on the other. 



192 



GEOMETRICAL PROBLEMS IX PROJECTION 



as, on rolling up, the edges must correspond, being the same 
initially. Join ei"W\ b''c'', etc., to complete the development. 
The proof, as before, Ues in the actual construction of the model 
and in showing that the cut is a plane surface. 

1112. Problem 7. To find the line of intersection of the 
surface of a right circular cone with a plane inclined to its axis. 

Construction. Let Fig. 190 show plan and elevation of the 
cone, and let T be the cutting plane. Through the axis, pass a 




Fig. 190. 



series of planes as ao, bo, co, etc. These planes cut rectilinear 
elements from the cone, showTi as o'a', o'b', o'c', etc., in the eleva- 
tion. At first, it is a good plan to dra^v the horizontal projection 
of the line of intersection, as many points are readily located 
thereon. For instance, e' is the point where the extreme ele- 
ment o'e' pierces the plane T, horizontally projected at e. Sim- 
ilarly, the actual element OD pierces the plane so that d and d' 
are corresponding projections. The points b and b' and the 
points a and a' are found in an identical manner, but, as in the 
case of the pyramid, the point c cannot be located in the same 
way. If the cone be revolved so that the element o'c' occupies 



INTERSECTIONS OF SURFACES BY PLANES 193 

the extreme position o's', then c' will be found at q' and c'q' 
will be the radius of the circle which the point c' describes; hence, 
make oc = c'q' and the resultant curve, which is an ellipse, may 
be drawn. 

The true line of intersection is shown in the supplementary 
view S. As in previous instances, first the axis ^'"q!" is drawn 
parallel to Tt', and equal in length to e'a' so that e"'e' and di!"di are 
both perpendicular to Tt^ From the axis, on a line from d', 
lay off d^'W = dm; also o'" c''' = oc and b''V=bn. A smooth 
curve then results in an ellipse. 

If accuracy is desired, it is better to lay off the major and 
minor axes of the ellipse and then draw the ellipse by other 
methods,* since great care must be used in this construction. 
The major axis is shown as ^'"o!" . To find the minor axis, 
bisect e'a' at u'and draw u'v', the trace of a plane perpendicular 
to the axis of the cone. This plane cuts the conical surface in a 
circle, of which wV is the radius. The minor axis is equal to 
the length of the chord of a circle whose radius is wV and whose 
distance from the centre of the circle is uV. 

1113. Problem 8. To find the developed surface in the 
preceding problem. 

Construction. The extreme visible element o's', Fig. 190, 
is shown in its true length and is the slant height of the cone. 
All elements are of the same length and hence, any indefinite 
arc AA, drawn so that o''A = o's' will be the first step in the 
development. The length of the arc AA is equal to the rectified 
length of the base, and, as such, is laid off and divided into any 
convenient number of parts. Eight equal parts are shown here 
because the auxiliary planes were chosen so as to cut the cone 
into eight equal parts. If the conical surface is cut along the 
element OA, then o'a' may be laid off on both sides equal to 
o''a'' in the development. The elements o'd', o'c' and o'b' 
are not shown in their true length, therefore, it is necessary to 
revolve the cone about the axis so as to make them parallel, 
in turn, to the vertical plane. The points will ultimately roach 
the positions indicated by p/ q' and r', and, hence, o"b" = o'r', 
o''c"=o'q' and o''d'' = oy. The element o'e' is shown in its 
true length and is laid off equal to o"e". A smooth curve through 
a"b"c''d", etc., completes the required development. 
* Sec Art. 000 in th's ronnoction. 



194 



GEOMETRICAL PROBLEMS IN PROJECTION 




When frustra of conical surfaces are to be developed the 
elements of the surface may be produced until they meet at the 
vertex. The development may then proceed 
along the usual hues. 

1114. Application of conical surfaces. 

As the ellipse is a sjTometrical curve, the upper 

and lower portions of the cone may be turned 

end for end so that the axes intersect. The 

resultant shape is indicated in Fig. 191, used at 

times in various sheet metal designs, such as 

oil-cans, tea-kettles, etc. 
Fig. 191. 

1115. Problem 9. To find the line of inter- 
section of a doubly curved surface of revolution with a plane 
inclined to its axis. 

Construction. Let Fig. 192 show the elevation of the given 
surface of revolution.* Atten- 
tion will be directed to the 
construction of the section 
shown in the plan. The high- 
est point on the curve is shoT^Ti 
at a, which is found from its 
corresponding projection a', at 
the point where the plane T 
cuts the meridian curve that 
is parallel to the plane of the 
paper. The point b is directly 
under b' and the length bb is 
equal to the chord of a circle 
whose radius is m'n' and 
whose distance from the cen- 
tre is b'n'. Similarly, cc is 
found by drawing an indefi- 
nite line under c' intersected 
by op = o'p' as a radius. As 

many points as are necessary are located, so as to get the true 
shape. One thing, however, must be observed: The plane XT 

* Many of these constructions can be carried to completion without 
actually showing the principal planes. The operations on them are per- 
formed intuitively. 




Fig. 192. 



INTERSECTIONS OF SURFACES BY PLANES 



195 



cuts the base of the surface of revolution at h' in the elevation 
and therefore hh is a straight line, which is the chord of a circle, 
whose radius is q'r' and which is located as shown in^the plan. 

The construction of the supplementary view resembles, in 
many respects, the construction in plan. The similar letters 
indicate the lengths that are equal to each other. 

1116. Problem 10. To find the line of intersection of a bell- 
surface with a plane. 

Construction. Fig. 193 shows the stub end of a connecting 
rod as used on a steam engine. The end is formed in the lathe 
by turning a bell-shaped surface of revolution on a bar of a 




Fig. 193. 



rectangular section, as shown in the end view, the shank being 
circular. The radius of curvature for the bell surface is located 
on the line ab as shown. The plane XT is tangent to the shank 
of the rod and begins to cut the bell-surface at points to the 
left of ab; hence, the starting point of the curve is at c. To 
find any point such as d, for instance, pass a plane through d 
perpendicular to the axis. It cuts the bell-surface in a circle 
(1024) whose radius is od" = o'd'". Whore the circle intersects 
the plane XT at d', project back to d, which is the required point 
on the curve. The scheme is merely this: By passing auxiliary 
planes perpendicular to the axis, circK^s iwv cut from the bell- 
surface which pierce the bounding ])lnn(>s in (lie nniuired ]K)ints 
of intersection. Attention might 1 c (hrcM'tcMl (o \\\o ])iont e 
which is on both ])lan and ehwation. Tliis is (rue Ix'cause tlu* 
planes XX and SS in((M-s(H'( in n line* which can pierce* (he \hA\- 



196 



GEOMETRICAL PROBLEMS IN PROJECTION 



surface at only one point. The manner in which nn = n'n' is 
located in the plan will be seen by the construction lines which 
are included in the illustration. To follow the description is 
more confusing than to follow the drawing. 

1117. Development by Triangulation. In the developments 
so far considered, only the cases of extreme simplicity were selected. 
To develop the surface an oblique cone or an oblique cylinder 
requires a slightly different mode of procedure than was used 




Fig. 194. 



heretofore. If the surface of the cone, for instance, be divided 
into a number of triangles of which the rectilinear elements form 
the sides and the rectified base forms the base, it is possible to 
plot these triangles, one by one, so as to make the total area 
of the triangles equal to the area of the conical surface to be 
developed. The method is simple and is readily applied in 
practice. Few illustrations will make matters clearer. 

1118. Problem 11. To develop the surfaces of an oblique 
hexagonal pjnramid. 

Construction. Fig. 194 shows the given oblique pyramid. 
It is first necessary to find the true lengths of all the elements of 
the surface. In the case of the pjTamid, the edges alone need 



INTERSECTIONS OF SURFACES BY PLANES 



197 



be considered. Suppose the pyramid is rotated about a perpen- 
dicular to the horizontal plane, through o, so that the base of 
the pyramid continually remains in its plane, then, when oa 
is parallel to the vertical plane, it is projected in its true length 
and is, hence, shown as the line o'di' . All the sides are thus 
brought parallel, in turn, to the plane and the true lengths are 
found. In most cases it will be found more convenient and less 
confusing to make a separate diagram to obtain the true lengths. 
To one with some experience, the actual lengths need not be drawn 
at all, but simply the distances a'', b'', c", etc., laid off. Then 
from any point o'" , draw arcs o'q!' = q"'^", oV = o''V, oT' = 




c" 6" 



Fig. 195. 



o'^'f , etc. From a'", lay off distances equal to the respective 
bases; in this case, ab = bc = cd, etc., and hence any one side 
will answer the purpose. Therefore, take that length on a 
divider and step off ab = a'"b"'=b'V = c'''d'" = a"'r', etc. 
The development is then completed by drawing the proper lines. 
The case selected shows the triangulation method applied 
to a surface whose sides are triangles. It is extremely simple 
for that reason, but its simplicity is still evident in the case of 
the cone as will be now shown. 

1119. Problem 12. To develop the surface of an oblique cone. 

Construction. Fig. 195 shows t lie giviMi ohliiiue cone. The 
horizontal projection of \\\v axis is op nnd the bast' is a circle, 



198 



GEOMETRICAL PROBLEMS IN PROJECTION 



hence, the cone is not a cone of revolution because the axis is 
incUned to the base. Revolve op to oq, parallel to the vertical 
plane. Divide the base into any number of parts ab, be, cd, 
etc. preferably equal, to save time in subsequent operations. 
The element oa is parallel to the plane, hence, o'a' is its true 
length. The element ob is not parallel to the plane, but can be 
made so by additional revolution as shown; hence o'b' is its 
true length. From any po'nt o", draw o'V = o'a', o''b'' = o'b', 
o''c" = o'c', etc. On these indefinite arcs step off distances a''b'' = 
h"c" = c"&", etc., equal to the rectified distances ab = bc = cd, 
etc. A smooth curve through the points a''b''c'', etc., will give 




c" 6'" 



Fig. 195. 



the development. As in previous problems, the cut is always 
made so as to make the shortest seam unless other requirements 
prevail. 

1120. Problem 13. To develop the surface of an oblique 
cylinder. 

Construction. In the case of the oblique cone, it was pos- 
sible to divide the surface into a series of triangles which were 
plotted, one by one, and thus the development followed by the 
addition of these individual triangles. In the case of the cylin- 
der, however, the application is somewhat different, although 
two cones may be used each of which has one base of the cylinder 



INTERSECTIONS OF SURFACES BY PLANES 



199 



as its base. The application of this is cumbersome, ani the 
better plan will be shown. 

Let Fig. 196 show the oblique cylinder, chosen, for convenience, 
with circular bases. Revolve the cylinder as shown, until it is 
parallel to the vertical plane. In the revolved position, pass a 
plane T through it, perpendicular to the axis. The curve so cut, 
when rectified, will develop into a straight line and the elements 
of the surface will be perpendicular to it. The true shape of the 
section of the cyhnder is shown by the curve a''b"c"d'', etc., 
and is an ellipse. Its construction is indicated in the diagram. 
Draw any base line AA and lay off on this the rectified portions 





a''''c'\^d'<e''Y''^g''y''h'\a:W]C 




Fig. 196. 



of the ellipse 2i"h" = ?i'"h"\ h"c" =h"'d" , etc. The revolved 
positions of the elements shown in the vertical plane are all 
given in their true sizes because they are parallel to that plane. 
Hence, lay off ?l"'o"' = di'o' , b''V=by, etc., below the base 
line. Do the same for the elements above the base line and the 
curve determined by the points so found will be the development 
of the given oblique cylinder. 

1121. Transition pieces. When an opening of one cross- 
section is to be connected with an opening of a different cross- 
section, th(.' connecting piece is called a transition piece. The 
case of transforming a circular cross-section to a square cross- 
section is quite common in heating and ventilating flues, boiler 
flues and the like. In all such cases, two possible nietliods offiT 



200 



GEOMETRICAL PROBLEMS IN PROJECTION 



themselves as solutions. The two surfaces may be comiected by 
a warped surface, the rectilinear elements of which are chosen so 
that the corners of the square are joined with the quarter points on 
the circle while the intermediate elements are distributed between 
them. A warped surface, however, is not developable; it is also 
difficult of representation and therefore commercially unsuited 
for application. The better method is to select some singly 
curved surface, or, a combination of planes and singly curved 




Fig. 197. 



surfaces since these are developable. The application, which 
is quite a common one, is shown in the following problem. 

1122. Problem 14. To develop the surface of a transition 
piece connecting a circular opening with a square opening. 

Construction. Let the two upper views of Fig. 197 represent 
the transition piece desired. The square opening is indicated 
by abed and the circular opening by efgh. The surface may 
be made up of four triangular faces aeb, bfc, cgd, dha, and four 
conical surfaces hae, ebf, fcg, gdh, whose vertices are at a, b, c 
and d. 

The development is shown at D, on the same figure. The 



INTERSECTIONS OF SURFACES BY PLANES 



201 



triangle a^'b^'e'' is first laid out so that the base a''b'' = ab; 
and the true length of the sides are determined as found in the 
construction leading to the position a'e'=a''e''=b''e''. The 
conical surfaces are developed by triangulation. For example, 
the arc he is divided into any number of parts em, mn, etc.; 
and the true lengths of the elements of the surface are found 





Fig. 198. 



by the construction leading up to the positions a'e', a'm', aV. 
Hence, a''e'' = a'e', a'W = aW; and m"e''=me is the rectified 
arc of the base of the conical surface. When four of these com- 
binations of triangle and conical surface are laid out, in their 
proper order, the development is then complete. 

If a rectangular opening is to be joined to an eUiptical open- 
ing, the manner in which 
this may be accomplished 
is shown in Fig. 198. 

1123. Problem 15. 
To develop the surface a 
transition piece connect- 
ing two elliptical openings 
whose major axes are at 
right angles to each other. 

Construction. Fig. 
199 shows three views of 
the elliptical transition 
piece. By reference to 
the diagram, it will be 
seen that the surface 
may be divided into 

eight conical surfaces, turned end for end; that is, four vertices 
are situated on one ellipse and four vertices are situated on the 




Fig. 199. 



202 



GEOMETRICAL PROBLEMS IX PROJECTION 



other. The shaded figure indicates the arrangement of the 

conical surfaces and the 
a/ elements are shown as 

shade Hnes. 
- In all developments of 

this character, the ar- 
rangement of the surfaces 
has considerable effect on 
the appearance of the 
transition piece when 
completed. In the case 
chosen, the eight conical 
surfaces give a pleasing 
result. If the intersec- 
tion of the eight alter- 
nate conical surfaces is 
objectionable, then it is 

possible to use a still larger number of di\'isions. 

The development of the conical surfaces is performed by the 

triangulation method. When finished, its appearance is that 




Fig. 199. 




\ 



/\ 



\ / \ / \ / 



b~- 



Fig. 200. 



of Fig. 200. The similar letters incUcate the order in which the 
siuiaces are assembled. 



1124. Development of doubh curved surfaces byapprox^ 
imation. Doubly curved surfaces are non-developable because 

successive elements cannot be made to coincide with a plane. It 
is possible, however, to di^dde the sm^ace into a series of singly 
curved surfaces and then to develop these. By di^-iding the 
doubly curved surface into a snfficiently large number of parts, 



INTEESECTIONS OF SURFACES BY PLANES 



203 



the approximation may be made to approach the surface as 
closely as desired. 

For doubly curved surfaces of revolution, two general methods 
are used. One method is to pass a series of meridian planes 
through the axis and then to adopt a singly curved surface 
whose contour is that of the surface of revolution at the 
meridian planes. This method is known as the gore method. 

The second method, known as the zone method, is to divide 
the surface into frusta of cones whose vertices lie on the axis 
of the doubly curved surface of revolution. The following 
examples will illustrate the application of the methods. 

1125. Problem 16. To develop the surface of a sphere by 
the gore method. 

Construction. Let the plan P of Fig. 201 show the sphere. 




Fig. 201. 



Pass a series of meridian planes aa, bb, cc, dd, through the sphere 
and then join ab, be, cd, etc. From the plan P construct the 
elevation E and then pass planes through i' and g', perpendicular 
to the axis of the sphere, intercepting equal arcs on the meridian 
circle, for convenience. Find the corresponding circles in the 
plan and inscribe an octagon (for this case) and detennine ef, 
the chord length for that position of the cutting ])lan(^s through 
{' and g'. This distance may tlu^n l)e laid off as e'f in the eleva- 
tion E and the curve n'e'a'e'm' be drawn. 

For the (level()i)in(Mit D, i( will b(MU)t(Hl that ab = a'b' = a"b" 



204 



GEOMETRICAL PROBLEMS IN PROJECTION 



and ef = e'f' = e"f'\ Also, the rectified distance m'g'=m"g", 
g'h'=g'V, hT=h'T', etc. A suitable number of points must 
be used in order to insure the proper degree of accuracy; the 
number chosen here is insufficient for practical purposes. By 
adding eight of these faces along the line a''b'', the development 
is completed. 

When this method is commercially applied, the gores may be 
stretched by hammering or pressing to their true shape. In 
this case they are singly curved surfaces no longer. 




Fig. 201. 



1126. Problem 17. To develop the surface of a sphere by 
the zone method. 

Construction. LetP, Fig. 202, be the plan and E, the eleva- 
tion of a sphere. Pass a series of planes a'a', b'b', c'c' etc., 
perpendicular to the axis of the sphere. Join a'b^ and pro- 
duce to h', the vertex of a conical surface giving rise to the frustum 
a'b'b'a'. Similarly, join b'c' and produce to g'. The last conical 
surface has its vertex e' in the circumscribing sphere. 

The development D is s milar to the development of any 
frustum of a conical surface of revolution. That is, with h" 
as a centre, draw an arc with h''a''=h'a' as a radius. At the 
centre line make a''o" = ao, o' V = on etc. The radius h''b" = h'b' 
so that the arc b''b''b'' is tangent internally, to the similarly 
lettered arc. The radius for this case is g''b"=g'=b'. The 
development is completed by continuing this process until all 
the conical surfaces are developed. 



INTERSECTIONS OF SURFACES BY PLANES 



205 





Fig. 202. 



1127. Problem 18. To develop a doubly curved surface of 
revolution by the gore method. Let ^_ 

Fig. 203 represent the doubly curved 
surface of revolution as finally approxi- 
mated. Through the axis at h pass a \ 
series of equally spaced meridian 
planes, and then draw the chords, one 
of which is lettered as dd. In the ele- 
vation E, pass a series of planes k'k', 
VVy m'm', perpendicular to the axis. 
The linos b'b', c'c', d'd', etc., are 
(hawn on the resulting singly curved 
surface and are shown in their true 
length. The corresponding projections 
of these lines in the ])lan P are also 
shown m their true leiigtli. 

To (leveloj) tli(> surface of one face lay olT \\\o r(H'liti(> 





.\\> 



206 



GEOMETRICAL PROBLEMS IX PROJECTION 



tance j'k' = j"k", k'r=k"r', etc. At 
^ etc.. lav off 





etc. 



the points j", k", 1", 
a'a' = a"a", b'b' = b"b", 

perpendicular to h"j" so that 



Fig. 203. 



the resulting figure D is s^iiimetrical 
alDout it. "When eight of these faces 
(or gores) are joined together and 
secured along the seams, the resulting 
figure ^ill be that shoTMi in plan and 
elevation. 

This surface may also be approxi- 
mated by the zone method. The 
choice of method is usually governed 
by conmiercial considerations. The 
gore method is perhaps the more 
economical in material. 



QUESTIONS OX CELIPTER XI 

1. State the general method of finding the piercing point of a line on 

a plane. 

2. Show by an obhque projection how the piercing points of a doubly 

cun'ed fine are found on the piincipal planes. 

3. Why is the projecting surface of a doubly cui'ved line a projecting 

cylinchical sui'face? 

4. What is a ''fine of intersection" of two surfaces? 

5. Distinguish between "line of intersection"" and ''section" of a solid. 

6. What is meant by the development of a sm-face? Explain fully. 

7. What are the essential characteristics of a developable surface? 

8. Why is a sphere non-developable? 

9. Why is any doubly curved surface non-developable? 

10. Why is a warped sm-face non-developable"? 

11. Prove the general case of finding the intersection of a right prism 

and a plane inclined to its axis. 

12. Show the general method of finding the development of the prism in 

Question 11. 

13. Prove the general case of finding the intersection of a right circular 

cylinder and a plane mchned to its axis. 
1-1. Show the general method of fincUng the development of the cyhnder 
in Question 13. 

15. When a right circular cyhnder is cut by a plane why must the ellipse 

be reversible? 

16. How is the torus approximated ^ith cyhndrical stufaces? 

17. Prove the general case of finding the intersection of a right p\Tamid 

and a plane inchned to its axis. 



INTERSECTIONS OF SURFACES BY PLANES 207 

18. Show the general method of finding the development of the right 

pyramid in Question 17. 

19. Prove the general case of finding the intersection of a right circular 

cone and a plane inchned to its axis. 

20. Show the general method of finding the development of the right 

cone in Question 19. 

21. Why is the elhpse, cut from a cone of revolution by an inchned plane, 

reversible? 

22. Prove the general case of finding the intersection of a doubly cur\^ed 

surface of revolution and a plane inchned to the axis. 

23. Is the surface in Question 22 developable? Why? 

24. Prove the general case of finding the intersection of a bell-surface 

with planes parallel to its axis. 

25. Is the bell-surface developable? Why? 

26. What is development by triangulation? 

27. Prove the general case of the development of the surface of an oblique 

pyramid. 

28. Prove the general case of the development of the surface of an oblique 

cone. 

29. Prove the general case of the development of the surface of an obhque 

cylinder. 

30. What is a transition piece? 

31. Why is it desirable to divide the surfaces of a transition piece into 

developable surfaces? 

32. Prove the general case of the development of a transition piece which 

joins a circular opening with a square opening. 

33. Prove the general case of the development of a transition piece 

which joins an elhptical opening with a rectangular opening. 

34. Prove the general case of the development of a transition piece 

which joins two elliptical openings whose major axes are at right 
angles to each other. 

35. What is the gore method of developing doubly curved surfaces? 

36. What is the zone method of developing doubly curved surfaces? 

37. Prove the general case of the development of a sphere by the gore 

method. 

38. Prove the general case of the development of a sphere by the zone 

method. 

39. Prove the general case of the development of a doul)ly curved sur- 

face of revolution by the gore method. 

40. A right octagonal prism has a circumscribing circle of 2" and 

is 4" high. It is cut by a plane, inclined 30° to the axis, 
and passes through its axis, I4" from the base. Find the 
section. 

41. Develop the surface of the i)risin of (Ju(^slion 40. 

42. A cylinder of n^vohition is 2]" in (liain(>ter and 3:,'" high. It is cut 

by a i)hine, inclined 45° to the axis and passes through its axis 
2" from tiic base. Find tlu^ section. 

43. Develop tlie surface oi the cylinder of (^)ueslion 12. 



208 GEOMETRICAL PROBLEMS IN PROJECTION 

44. A right octagonal pjTamid has a circumscribing base circle of 2|" 

and is 4" high. It is cut by a plane, inchned 30° to the axis and 
passes through its axis 2" from the base. Find the section. 

45. Develop the surface of the pyramid of Question 44. 

46. A right circular cone has a base 2" in diameter and is 3f" high. 

It is cut by plane inchned 45° to the axis and passes through its 
axis 2^' from the base. Find the section. 

47. Develop the cone of Question 46. 

48. A 90° stove pipe elbow is to be made from cyhnders M' in diameter. 

The radius of the bend is to be 18". Divide elbow into 8 parts. 
The tangent distance beyond the quadrant is 4'\ Develop the 
surface to suitable scale. 

49. Assume a design of a vase (a doubly curved surface of revolution) 

and make a section of it. 

50. The stub end of a connecting rod of a steam engine is 4"x6"; the 

rod diameter is Z\" . The bell-surface has a radius of 12". Find 
the lines of intersection. Draw to suitable scale. 

51. An obhque pjTamid has a regular hexagon for a base. The circum- 

scribing circle for the base has a diameter of 2"; the altitude is 
3^" and the projection of the apex is 2\" from the centre of the 
base. Develop the surface of the pyramid. 

52. An obhque cone has a circular base of 2\^' in diameter and an altitude 

of 3". The projection of the vertex on the plane of the base is 
Ij" from the centre of the base. Develop the conical surface. 

53. An obhque cyhnder has a circular base \\" in diameter; it is 2\" 

high and the inclination of the axis is 30° "with the base. Develop 
the surface. 

54. A transition piece is to be made joining a 3'-0"x5'-6" opening 

with a 4'-0" diameter circle. The distance between openings 
3-6". Develop the surface. Use a suitable scale for the drawing. 

55. A transition piece is to be made joining an opening ha\drig a 3'-0" 

X4'-0" opening to a 2'-0"x6'-0" opening (both rectangular). 
The distance between openings is 4-0". Develop the sheet and 
use suitable scale for the drawing. 

56. A square opening 4"-0"x4'-0" is to be joined "with an elhptical 

opening whose major and minor axes are 5'-0"X3'— 0" respec- 
tively. The distance between openings is 3'-0". Develop the 
sheet. Use suitable scale in making the drawing. 

57. A circular opening having a diameter of 3-0" is to be connected 

with an elliptical opening whose major and minor axes are 
4'-0"x2'-6" respectively. The distance between openings is 
6'-0". Develop the sheet. Use suitable scale in making the 
drawing. 

58. An opening ha^ing two parallel sides and two semicircular ends has 

overall dimensions of 3'-0"x5'-6". This opening is to be joined 
with a similar opening ha\ing dimensions of 3 -6" X 4^-6". The 
distance between openings is 3'-6". Develop the sheet. Use 
suitable scale in making the drawing. 



INTERSECTIONS OF SURFACES BY PLANES 209 

59. A frustum of a cone has a base which is made up of two parallel 

sides and two semicircular ends and whose overall dimensions are 
S'-6" x6'-0". The height of the vertex above the base is 8-6"; 
that of the upper base is 4'-0" above the lower base. Develop 
the sheet and use suitable scale in making the drawing. 

60. An eUiptical opening having a major and minor axis of 5'-0" and 

4'-0", respectively, is to be joined with a similar opening but 
whose major axis is at a right angle. Develop the sheet and use 
suitable scale in making the drawing. 

61. An elliptical opening having a major and minor axis of 4-6" and 

S'-Q", respectively, is to be joined with a similar opening but 
whose major axis is at an angle of 30°. Develop the sheet and use 
suitable scale in making the drawing. 

62. An elliptical opening having a major and minor axis of S'-O" and 

3'-6", respectively, is to be joined with another elhptical opening 
whose major and minor axes are 4'-6''' and 3'-6", respectively. 
Develop the sheet and use suitable scale in making the drawing. 

63. A sphere is 6" in diameter. Develop the surface by the gore 

method and divide the entire surface into sixteen parts. Use a 
suitable scale in making the drawing. 

64. A sphere is 6" in diameter. Develop the surface bj^ the zone 

method and divide the entire surface into twelve parts. Use a 
suitable scale in making the drawing. 

65. Assume some doubly curved surface of revolution and develop the 

surface by the gore method. 

66. Assume some doubly curved surface of revolution and develop the 

surface by the zone method. 






1 



CHAPTER XII 

INTERSECTIONS OF SURFACES WITH EACH OTHER, AND THEIR 

DEVELOPMENT 

1201. Introductory. When two surfaces are so situated 
vdth. respect to each other that they intersect, they do so in a 
Une which is called the line of intersection. It is highly desirable 
in the conception of intersections to realize that certain elements 
of one surface intersect certain elements on the other surface, 
and that the locus of these intersecting elements is the desired 
line of intersection. 

The line of intersection is found by passing auxiliary surfaces 
through the given surfaces. Lines will thus be cut from the given 
surfaces by the auxihary surfaces, the intersections of which 
yield points on the desired line of intersection. The simplest 
auxihary surface is naturally the plane, but sometimes cylindrical,* 
conical and spherical surfaces may find application. It is not 
necessary only to use a simple type of auxiliary surface, but 
also to make the surface pass through the given surfaces so as 
to cut them in easily determinable lines — preferably, in the 
elements. As illustrations of the latter, the case of a concial 
surface cut by a plane passing through the vertex furnishes an 
example. It cuts the surface in straight lines or rectilinear 
elements. Likewise, a plane passed through a cylindrical sur- 
face parallel to any element will cut from it, one or more lines 
which are also rectiliner elements. For doubly curved surfaces 
of revolution, a plane passed through the axis (a meridian plane) 
cuts it in a meridian curve. When the plane is perpendicular 
to the axis, it cuts it in circles. An interesting and valuable 

* A careful distinction must be made between solids and their boimding 
surfaces. The terms, cyhnder and cyhndrical surface are often used indis- 
criminately. The associated idea is usually obtained from the nature of the 
problem. 

210 



INTERSECTIONS OF SURFACES WITH EACH OTHER 211 

property of spheres is that all plane sections in any direction, 
or in any place, are all circles, whose radii are readily obtain- 
able. 

When prisms or pyramids intersect, auxiliary planes are 
not required to find their intersection, because the plane faces and 
the edges furnish sufficient material with which to accomplish 
the desired result. These types of surfaces are therefore exempt 
from the foregoing general method. 



1202. Problem 1. To find the line of intersection of the 
surfaces of two prisms. 

Construction. Let Fig. 204 represent the two prisms, one 
of which, for convenience, is a right 
hexagonal prism. The face ABDC 
intersects the right prism in the line 
nm, one line of the required intersec- 
tion. The -face CDFE is intersected 
by the edge of the right prism in the 
point o, found as shown in the con- 
struction. Hence, mo is the next line 
of the required intersection. Similarly, 
the edge EF pierces the faces of the 
right prism at p, and op is the con- 
tinuation of the intersection. In this 
way, all points are found. The curve 
must be closed because the inter- 
penetration is complete. Only the 

one end where the prism enters is shown in construction. 
The curve where it again emerges is found in an identical 
manner. 




ak c t eO 



Fig. 204. 



1203. Problem 2. To find the developments in the preceding 
problem. 

Construction. The development of the right prism is 
quite simple and should be understood from the preceding 
chapter. The points at which the oblicjue prism inter- 
sects the right prism are also shown. Thus, it is only 
necessary to make sure on what ehMuent the point pierces, 
and this may be obtained from the plan viinv of tho right 
prism. 



212 



GEOMETRICAL PROBLEMS IN PROJECTION 



In developing the oblique prism, D' of Fig. 205, the same 
general scheme is followed as was used in developing the oblique 
cylinder (1120). Revolve the prism of Fig. 204 so that the 
elements are parallel to the plane of projection; they are then 
shown in their true length. Pass a plane perpendicular to the 
edges and this line of intersection will develop into a straight line 
which is used as a base line. Find now the true section of the 
prism and lay off the sides perpendicular to the base line and at 
their proper distances apart. Lay off the lengths of the edges 
above and below the base line and join the points by lines to 
complete the development. 

When actually constructing this as a model, it will be well to 




Fig. 205. 



carry out the work as shown, and make the oblique prism in 
one piece. It may then be inserted in the opening provided for 
in the right prism, as indicated. 

1204. Problem 3. To find the line of intersection of two cylin- 
drical surfaces of revolution whose axes intersect at a right angle. 

Construction. Let Fig. 206, represent the given cylindrical 
surfaces. Through o', pass a series of planes a' a', b'b', c'c', 

etc. The elements cut from the cylindrical surface by these 
planes interest the other cylindrical surface at the points a, e, b, d, c. 
The two upper views lead to the construct on of the lower view. 
It will be seen by this construction that the entire scheme of 
locating points on the curve lies in the finding of the successive 
intersections of the elements of the surfaces. 

If the cylindrical surfaces have elliptical sections, instead of 



INTERSECTIONS OF SURFACES WITH EACH OTHER 213 



circular sections as shown, the method of procedure will be found 
to be much the same. 

1205. Problem 4. To find the developments in the preceding 
problem. 

Construction. As the elements of the surfaces depicted in 
Fig. 206 are parallel to the plane of the paper, they are therefore, 
shown in their true length. Hence, to develop the surfaces, 

i i i i iii^i I i 




Fig. 206. 



Fig. 207. 



rectify the bases and set off the points at proper distances from 
the base line and also their rectified distance apart. The 
appearance of the completed developments are shown in Fig. 207. 
This problem is similar to the development of the steam 
dome on a locomotive boiler. 



1206. Problem 5. To find the line of 
intersection of two cylindrical surfaces of 
revolution whose axes intersect at any angle. 

Construction. Fig. 208 shows the two 
cylindrical surfaces chosen. If a series of 
auxiliary planes be passed parallel to the 
plane of the intersecting axes they will cut 
the cylindrical surfaces in rectilinear ele- 
ments. The elements of one surface will 
intersect the elements of the other surface 
in the required points of the line of inter- 
section. The construction shown in the 
figure should be clear from the similar lettering 
l)oints on th(^ line of intersection. 




imilar 



214 



GEOMETRICAL PROBLEMS IX PROJECTION 



It may be desirable to note that this method of locating 
the required points on the curve is also applicable to the con- 
struction of Fig. 206, and vice versa. 

If the plane of the intersecting axes is not parallel to the plane 
of the paper, the construction can be simplified by revolving the 
cylindrical surfaces until they become parallel. The method 
of procedure is then the same as that given here. If the axes 
cannot be made to lie in the same plane, then the construction 
is more difficult. The method in such cases is to pass a plane 
through one cylinder so as to cut in in elements; and the same 
plane will cut the other cylinder in some line of intersection. 

The intersection of;'these will 
yield points on the desired line 
of intersection. 

1207. Problem 6. To find 
the developments in the pre- 
ceding problem. 

Construction. Fig. 209 
shows the development as it 
appears. If the plane of the 
base is perpendicular to the 
elements of the surface, the 
base will develop into a straight 
line. The elements will be at 
Fig, 209. ^i§^^ angles to the base fine 

and as they are shown in 
their true length, they may be laid off directly from Fig. 208. 




1 ^ 


a" 


1 




r 


S 


l\i 






i 






D' 


^'1 




\d' 




iH-i 


ri 


f^^^' 

^ 







Fig. 210. 



Fig. 211. 



1208. Application of intersecting cylindrical surfaces to 
pipes. The construction of pipe fittings as commercially used 
furnish examples of intersecting cj^hndrical surfaces. Fig. 210 



INTERSECTIONS OF SURFACES WITH EACH OTHER 215 

shows two cylindrical surfaces whose diameters are the same and 
whose axes intersect at a right angle. The Une of intersection 
for this case will be seen to consist of two straight lines at right 
angles to each other. In Fig. 211, the cylindrical surfaces have 
their axes intersecting at an angle of 45° and have different 
diameters. 

1209. Problem 7. To find the line of intersection of two 
cylindrical surfaces whose axes do not intersect. 

Construction. Fig. 212 shows two circular cylindrical sur- 
faces which are perpendicular to each other and whose axes are 
offset. Pass a series of planes ab, cd, ef, etc., through o. These 
planes cut the cylindrical surface in elements, which in turn, 
intersect elements of the other surface. As the construction 
lines are completely shown, the description is unnecessary. 





J— 


1 — \ — 


4; 


\a"' 




^ 


4^ 


I 

i 



Fig. 212. 



Fig. 213. 



1210. Problem 8. To find the developments in the pre- 
ceding problemo 

Construction. First consider the larger cylindrical surface, 
D', Fig. 213, which is cut along any clement. With a divider, 
space off the rectified distances between elements; and on the 
proper elements, lay off W to correspond with b'' ; k'^'d'" = k"d" 
r'T'' = i'T', etc. 

For the smaller cylindrical surface, rectify the entir(^ circle 
ace ... la and divide^ into iho numlxM' of ])arts shown. As the 
elements avv shown i)arall(4 to the ])lan(^ of ])r{)j(H'tion in the 
side elevation, they may be (linM'lly ])l()t(iMl ns indicattHl by (ht^ 

This {'()inph'l(\s Ww final (h'V(^io|)nuMit . 



curve g'"i'"k'" 



e'"g'". 



216 GEOMETKICAL PROBLEMS IN PROJECTION 

1211. Intersection of conical surfaces. If a plane be passed 
through the vertex of a conical surface it cuts it in rectilinear 
elements if at all. When two conical surfaces intersect it is 
possible to pass a plane through the vertices of both. Thus, 
the plane may be made to intersect both surfaces in rectilinear 
elements, the intersections of which determine points on the 
required line of intersection. 

Fig. 214 shows this pictorially. If the vertices m and n be 
joined by a line and its piercing point o be found, then, any line 
through o will determine a plane. Also, if od be the line chosen, 
the plane of the lines mo and od will cut the conical surfaces in 




Fig. 214. 

the elements ma, mb, nc and nd. These elements determine 
the four points e, f, g, h. Any other line through o will, if prop- 
erly chosen, determine other elements which again yield new 
points on the line of intersection. The following problem will 
bring out the details more fully. 

1212. Problem 9. To find the line of intersection of the 
surfaces of two cones, whose bases may be made to lie in the 
same plane, and whose altitudes differ. 

Construction. Let A and B, Fig. 215 be the vertices of the 
two cones in question, whose bases are in the horizontal plane. 
If the bases do not lie in the principal planes, a new set of prin- 
cipal planes may be substituted to attain the result. Join A 
and B by a line and find where this line pierces the horizontal 



INTERSECTIONS OF SURFACES WITH EACH OTHER 217 

plane at c. Any line, through c, lying in the horizontal plane, 
in addition to the line AB will determine a plane, which may 
cut the conical surfaces in elements. Let, for instance, cd be 
such a line. This line cuts the bases at d, e and f, the points 
which will be considered. The elements in the horizontal pro- 
jection are shown as db, eb and fa; in the vertical projection, 
they are d'b^ e'b' and fa'. The element fa' intersects the element 
d'b' at g', and the element e'b' at h', thus determining g' and 
h', two points on the vertical projection of the required line of 
intersection. The corresponding projections g and h may also 
be found, which determine points on the horizontal projection 




Fig. 215. 



of the same line of intersection. Consider, now, the line ci, 
tangent to the one cone. Its element is ib in the horizontal 
projection and i'b' in the vertical projection; the corresponding 
element cut from the other cone is ja in the horizontal j^rojoc- 
tion and j'a' in the vertical projection. The })oints k and k' 
are thus found, but they represent only one point K on the actual 
cones. 

This process is continued until a sufficient number of points 
are determined, so that a smooth curve can be drawn through 
them. The cones chosc^n, hav(^ com])lete intiTponotration, a^ 
one cone goes entirely through the other. Thus, there are two 
distinct curves of intersection. That, near the apex B, is found 
in an identical way with the prcccHJing. 



218 



GEOMETRICAL PROBLEMS IN PROJECTION 



Attention might profitably be called to the manner in which 
points are located so as to miminize confusion as much as possible. 
To do this, draw only one element on each cone at a time, locat- 
ing one, or two points, as the case may be; then erase the con- 
struction lines when satisfied of the accuracy. Several of the 
prominent points of the curve may be thus located and others 
estimated if considerable accuracy is not a prerequisite. If 
the surfaces are to be developed and constructed subsequently, 
however, more points will have to be established. 

There is nothing new in the development of these cones, the 
case is similar to that of any oblique cone and is therefore omitted. 
One thing may be mentioned in passing however, and that is, 
while drawing the elements to determine the contour of the base, 
the same elements should be used for locating the line of inter- 
section as thereby considerable time is saved. 



1213. Problem 10. To find the line of intersection of the 
surfaces of two cones, whose bases may be made to lie in the 
same plane and whose altitudes are equal. 

Construction. Let A and B, Fig. 216, be the desired cones. 

Join A and B by a line which is 
chosen, parallel to the planes of 
projection. Hence, this line cannot 
pierce the plane of the bases, and 
the preceding method of finding 
the line of intersection is thus 
inapplicable. It is possible, how- 
ever to draw a line cd parallel to 
ab. These lines therefore determine 
a plane which passes through both 
vertices and intersects the surface in 
rectilinear elements. The comple- 
tion of the construction becomes 
evident when the process which 
leads to the finding of F and G is understood. 

The cases of intersecting cones so far considered have been 
so situated as to have bases in a common plane. This may 
not always be convenient. Therefore to complete the subject, 
and to cover emergencies, an additional construction will be 
studied. 




Fig. 216. 



INTERSECTIONS OF SURFACES WITH EACH OTHER 219 

1214. Problem 11. To find the line of intersection of the 
surfaces of two cones whose bases lie in different planes. 

Construction. Let A and B, Fig. 217, be the vertices of the 
two cones. The cone A has its base in the horizontal plane; B 
has its base in the plane T, which is perpendicular to the vertical 
plane. Join the vertices A and B by a line, which pierces the 
horizontal plane at c and the plane T in the point d' horizontally 
projected at d. Revolve the plane T into the vertical plane, 
and, hence, the trace Tt becomes Tt". The angle t'Tt" must 




Fk;. 217. 



be a right angle, because it is perpendicular to the vertical plane, 
and, therefore cuts a right angU* from tlu^ principal ])laues. The 
piercing point of the line AB is at d" in the revolved position, 
where d'd" is ecjual to tlie distance of d from tlu^ ground line. 
In the revolved position of the plane T, draw the base of the 
cone B and from it accurately construct the horizontal projec- 
tion of the base. Draw a line d"e" tangiuit to the revolved 
position of the base of th(^ cone B at f. From it, find f, the 
horizontal, and f, the vertical projection of this point, and draw 
the elcMuents fb and f'b'. Lay olT Te"=Te, and draw ce. The 
lines CD and DE detiM'mine a plane which cuts the horizontal 



220 



GEOMETRICAL PROBLEMS IN PROJECTION 



projection of the base of the cone A in g and h, and thus, also, 
the two elements ag and ah which are horizontal projections, 
and a'g' and a'h' the corresponding vertical projections of the 
elements. The element BF intersects the elements AG and AH 
in the points M and N shown, as usual, by their projections. 
M and N are therefore two points of the required curve, being 
prominent points because the elements are tangent at these points. 
To obtain other points, draw a line d"i" and make Ti" =Ti. ^Miere 
ci cuts the cone A, draw the elements as sho^Ti, also the correspond- 




1 



Fig. 217. 

ing elements of the cone B. The points of the curve of intersec- 
tion are therefore as indicated. 

One fact is to be observed in this and in similar constructions. 
Auxihary planes are passed through the vertices of both cones, 
cutting, therefore, elements of the cones whose intersection 
determine points on the curve. All planes through the vertices 
of both cones must pass through c in the horizontal plane and 
through d" in the plane T. Also, such distances as Te" must 
equal Te because these auxiliary planes cut the horizontal plane 
and the plane T on their line of intersection Tt; the revolution 
of the plane T into the vertical plane does not disturb the location 




INTERSECTIONS OF SURFACES WITH EACH OTHER 221 

of any point on it. Hence such distance as Te will be revolved 
toTe" where Te = Te''. 

When applying this problem to a practical case, it would be 
better to select a profile plane for the plane T, as then Tt" would 
coincide with the ground line and all construction would be 
simplified. It has not been done in this instance, in order to 
show the generality of the method, and its adaptation to any 
condition. 

1215. Types of lines of intersection for surfaces of cones. 

When the surfaces of two cones are situated so that there is com- 




FiG. 218. Fig. 219. 

plete interpenetration, the line of intersection will appear as 
two distinct closed curves. Fig. 215 is an example. 

If the surfaces of the cones are such that the interpenetration is 
incomplete, only one closed curve will result. Figs. 2U) and 217 
are examples of this cas(\ 

When the surfaces of two cones have a common tangent plane 
then the curve is closed and crosses itself once. An example 
under this heading is given in Fig. 218. 

It is possible to have two cones the surfaces of which have 
two common tangent planes. In this case there iwv two closed 
curves wliich cross each other twice. This illustration ai)pears 
in Fig. 219. 



222 



GEOMETRICAL PROBLEMS IN PROJECTION 



1216. Problem 12. To find the line of intersection of the 
surfaces of a cone and a cylinder of revolution when their axes 
intersect at a right angle. 

Construction. Let Fig. 220 represent the cone and the 
cylinder. Through the cone, pass a series of planes perpendicular 
to its axis. If the planes are properly chosen, they will cut the 
conical surface in circles and the cylindrical surface in rectilinear 
elements, the intersection of which determine points on the line 
of intersection. A reference to Fig. 220 will show this in con- 
struction. As a check on the accuracy of the points on the curve, 
it is possible to draw elements of the conical surface through some 
point; the corresponding projections of the elements must con- 




FiG. 220. 



tain the corresponding projections of the points. In the 
illustration, the elements OA and OB are drawn through the 
points E and J respectively. 

1217. Problem 13. To find the line of intersection of the 
surfaces of a cone and a cylinder of revolution when their axes 
intersect at any angle. 

Construction. The given data is shown in Fig. 221. In 
this case it is inconvenient to pass planes perpendicular to the 
axis of the cone, since ellipses will be cut from the cylinder. A 
better plan is to find m, the intersection of their axes, and use 
this as a centre for auxiliary spherical surfaces. The spherical 
surfaces intersect the surfaces of revolution in circles (1025). 



INTERSECTIONS OF SURFACES WITH EACH OTHER 223 

The details of the construction are shown in Fig. 221. To 
check the accuracy of the construction it is possible to draw an 
element of one surface through some point and then find the 
corresponding projection of the element; the corresponding 
projection of the point must be located on the corresponding 
projection of the element. Two elements OF and OG are showTi 
in the figure. The points determined thereby are shown at D. 

1218. Problem 14. To find the line of intersection of the 
surfaces of an oblique cone and a right cylinder. 

Construction. Let Fig. 222 illustrate the conditions assumed. 




0"^^ 




f" 


'^ 


^^"^■^^^^^[y 


9' 




Fig. 222. 



To avoid too many construction lines, the figures have been 
assumed in their simplest forms. Pass any plane through o, 
perpendicular to the horizontal plane; it cuts elements DO and 
CO from the cone and the element from the cylinder which is 
horizontally projected at f and vertically at f'g'. In the* vertical 
projection, the elements c'o' and d'o' intersect the element f'g' 
at g' and h', two points of the nniuired curve. The extreme 
element OE determines the ]K)inl i l)y \ho same method. There 
are two distinct lines of intersection, in this case, due to comph^te 
interpenetration. The points on the hn(» of intiTsiM'tion near tlie 
vertex are h)ca(ed in a manner similar to that shown. 



224 



GEOMETRICAL PROBLEMS IN PROJECTION 



1219. Problem 15. To find the developments in the pre- 
ceding problem. 

Construction. The oblique conical surface is developed 
by triangulation. In Figs. 222 and 223, the lines o'q! and o'b' 
are shown in their true length in the vertical projection; hence, 
lay off o'a' = o''a'', and o'b' = o^'b''. If the element oe be revolved 
so that it is parallel to the vertical plane, the point i', in the vertical 
projection, will not change its distance above the horizontal 
plane during the revolution. Hence it moves from i' to k', the 
revolved position. Accordingly, lay off o"i''=o'k' and one 
point of the intersection on the development is obtained. Other 
points, of course, are found in absolutely the same manner. 




I 



Fig. 223. 



Extreme accuracy is required in most of these problems. Con- 
structions like this should be laid out to as large a scale as con- 
venient. The development of the cy Under is also shown and is 
perhaps clear without explanation. 

1220. Problem 16. To find the line of intersection of the 
surfaces of an oblique cone and a sphere. 

Construction. Let Fig. 224 represent the cone and sphere 
in question. The general scheme is to pass planes through the 
vertex of the cone perpendicular to the horizontal plane. These 
planes cut the cone in rectilinear elements and the sphere in 
circles; the intersection of the elements and the circles so cut 
will determine points on the curve. 

Thus, through o, draw a plane which cuts the cone in a and b 
and the sphere in d and e. The elements cut are shown as o'a' 



INTERSECTIONS OF SUKFACES WITH EACH OTHER 225 




and o'b' in the vertical projection. Attention will be confined 
to the determination of g', one point 
on the lower curve, situated on the 
element OA. The circle cut from the 
sphere by the plane through oa and 
ob has a diameter equal to de. If pc 
is a perpendicular to de from p the 
centre of the sphere, then c is the hori- 
zontal projection of the centre of the 
circle de, and cd and ce are equal. 
If the cutting plane is revolved about 
a perpendicular through o to the hori- 
zontal plane, until it is parallel to the 
vertical plane, a will move to a." and 
o'a''' will be the revolved position of 
this element. The centre of the circle _ 
will go to h in the horizontal projec- 
tion and h' in the vertical projection, 
because, in this latter case, the distance 
of h' above the horizontal plane does 
not change in the revolution. The 

element and the centre of the circle in a plane parallel to the 
vertical plane are thus determined. Hence, with cd as a radius 
and h' as a centre, describe an arc, cutting o'a''' in f . On counter 

revolution, f goes to g' on o'a', the 
original position of the element. The 
point g' is therefore one point on 
the curve. Every other point is 
found in the same wa}'. 

1221. Problem 17. To find the 
line of intersection of the surfaces 
of a cylinder and a sphere. 

Construction. Fig. 225 pic- 
tures the condition. Pass a series of 
pianos, through (lie cylinder and tlie 
sphere, perpendicular to the plane of tlie base of the cylinder. One 
position of the cutting plane cuts the cylinder in a'b' and the 
sphere in c'd'. Construct a su])plenientnry \'w\\ S with thi^ centre 
of the sphere at o" as shown. The (^hMncnts api)ear ns a"a" and 
b''b''. The diameter of (ho circK^ cut from tlie sphere isc'd'and 



Fig. 224. 




Fig. 225. 



226 



GEOMETRICAL PROBLEMS IN PROJECTION 



with c'e' as a radius, (equal to one-half of c'd'), draw an arc cutting 
the elements at f'g''h" and i". Lay off f 'j" = fj; g"j" = gj, etc., 
and the four points f, g, h, i are determined on the required 
view. These points are on the required line of intersection. 

1222. Problem 18. To find the line of intersection of two 
doubly curved surfaces of revolution whose axes intersect. 

Construction. Let Fig. 226 represent the surfaces in ques- 
tion. With o, the intersection of the axes, as a centre, draw 
a series of auxiliary spherical surfaces. One of these spherical 
surfaces cuts the surface whose axis is on in a circle whose diameter 
is cd. This same auxihary sphere cuts the surface whose axis is 





Fig. 226. 



om in a circle projected as ef. Hence cd and ef intersect at a, 
a point on the required line of intersection. 

In the view on the right, only one of the surfaces is shown. 
The location of the corresponding projections of the line of 
intersection will be evident from Art. 1027. 

When the axes do not intersect then the general method is 
to pass planes so as to cut circles from one surface of revolution 
and a curve from the other. The intersections determine points 
on the curve. It is desirable in this connection to chose an arrange- 
ment that gives the least trouble. Xo general method can be 
given for the mode of procedure. 

1223. Commercial application of methods. In practice, 
it is always desirable as a matter of time to turn the objects so 
that the auxiliary surfaces may be passed through them ^-ith 
the least effort. Frequently, many constructions may be car- 
ried out without any special reference to the principal planes. 



INTERSECTIONS OF SURFACES WITH EACH OTHER 227 

There is no harm in omitting the principal planes, but the student 
should not go to the extreme in this omission. With the 
principal planes at hand, the operations assume a familiar form, 
which will have a tendency to refresh the memory as to the basic 
principles involved. All the operations in the entire subject 
have a remarkable simplicity in the abstract; the confusion 
that sometimes arises is due not to the principles involved, but 
solely to the number of construction lines required. It, therefore, 
seems proper to use such methods as will lead to the least con- 
fusion, but it should always be borne in mind that accuracy is 
important at all hazards. 

QUESTIONS ON CHAPTER XII 

1. State the general method of finding the intersection of any two 

surfaces. 

2. When the surfaces are those of prisms or pyramids, is it necessary 

to use auxiliar}^ planes as cutting planes? Why? 

3. Prove the general case of finding the fine of intersection of the sur- 

faces of two prisms. 

4. Show the general method of finding the development of the prisms 

in Question 3. 

5. Prove the general case of finding the line of intersection of two 

cylindrical surfaces of revolution whose axes intersect at a right 
angle. 

6. Show the general method of finding the development of the surfaces 

in Question 5. 

7. Prove the general case of finding the line of intersection of two 

cylindrical surfaces of revolution whose axes intersect at any 
angle. 

8. Show the general method of finding the development of the surfaces 

in Question 7. 

9. Prove the general case of finding the line of intersection of two 

cylindrical surfaces whose axes do not intersect. 

10. Show the general method of finding the development of the cylin- 

drical surfaces in (Question 9. 

11. State the general method of finding the line of intersection of two 

conical surfaces. 

12. Prove the g(Mieral cas(^ of finding the line of intcM-section of tlu^ sur- 

faces of two ()l)li(iue cones whose bases may he made to l)e in 
the same plan(> and wliose altitudes difler. 

13. Show the general metliod of fin(liii<;- the d(^vel()i)iuents of the surf;i('(\s 

of the cones in (Question 12. 

14. Prove tlie general case of finding the line of intersection of the sur- 

faces of two ()l)li(iue cones wli()S(> hases may he made to lie in the 
same plane and whose altitudes are e(iual. 



228 GEOMETRICAL PHOBLEM.- IN PROJECTION 

15. Show the general method of finding the developments of the 

surfaces of the cones m Question 14. 

16. Prove the general case of finding the fine of intersection of the sur- 

faces of two obUque cones who.se bases He in different planes. 

17. Show the general method of finding the developments of the cones 

in Question 16. 
IS. When the surfaces of two cones have complete interpenetration, 
discuss the nature of the line of intersection. 

19. When the surfaces of two cones have incomplete penetration, discuss 

the nature of the fine of intersection. 

20. ^^^len the surfaces of two cones have a common tangent plane, discuss 

the nature of the line of intersection. 

21. When the surfaces of two cones have two common tangent planes, 

discuss the nature of the line of iatersection. 

22. Prove the general case of finding the line of intersection of the sur- 

faces of a cone and a cylinder of revolution when their axes inter- 
sect at a right angle. 

23. Show the general method of finding the developments of the surfaces 

ID Question 22. 

24. Prove the general case of finding the line of intersection of the sur- 

faces of a cone and a cyhnder of revolution when their axes inter- 
sect at any angle. 

25. Show the general method of finding the developments of the sur- 

faces in Question 24. 

26. Prove the general case of finding the line of intersection of the sur- 

faces of an oblique cone and a right cylinder. 

27. Show the general method of finding the developments in Question 26. 
2S. Prove the general case of finding the fine of intersection of the sur- 
faces of an oblique cone and a sphere. 

29. Show the general method of finchng the development of the cone in 

Question 2S. 

30. Develop the stuface of the sphere in Question 2S by the gore 

method. 

31. Prove the general case of finchng the fine of intersection of the sur- 

faces of a sphere and a c^-finder. 

32. Show the general method of finding the development of the cylinder 

in Question 31. 

33. Develop the sin^ace of the sphere in Question 31 by the zone method. 
M. Prove the general case of finding the fine of intersection of two 

doubly cur^'ed sm^aces of revolution whose axes intersect. 

35. Develop the surfaces of Question 34 by the gore method. 

36. Develop the surfaces of Question 34 by the zone method. 

37. State the general method of finding the fine of intersection of two 

doubly cur^^ed surfaces of revolution whose axes do not intersect. 

38. What items are to be considered when apphing the principles of 

intersections and developments to commercial problems? 

39. Is it always necessary- to use the principal planes when sohing prob- 

lems relating to intersections and developments? 




INTERSECTIONS OF SURFACES WITH EACH OTHER 229 

40. Find the intersection of the two prisms shown in Fig. 12-A. Assume 

suitable dimensions. 

41. Develop the prisms of Question 40. 

42. Two cyhnders of 2" diameter intersect at a right angle. Find the 

hne of intersection of the surfaces. Assume 
suitable dimensions for their lengths and position 
with respect to each other. 

43. Develop the surfaces of Question 42. 

44. Two cyhnders of 2" diameter and If diameter 

intersect at a right angle. Find the hne of 
intersection of the surfaces. Assume suitable 
dimensions for their lengths and position \sith 
respect to each other. 

45. Develop the surfaces of Question 44. 

46. Two cyhnders of 2" diameter intersect at an angle p^^ I'^-A. 

of 60°. Find the hne of intersection of the sur- 
faces. Assume suitable dimensions for their lengths and position 
with respect to each other. 

47. Develop the surfaces of Question 46. 

48. Two cyhnders of 2" diameter intersect at an angle of 45°. Find 

the line of intersection of the surfaces. Assume suitable dimen- 
sions for their lengths and position with respect to each other. 

49. Develop the surfaces of Question 48. 

50. Two cyhnders of 2" diameter intersect at an angle of 30°. Find 

the hne of intersection of the surfaces. Assume suitable dimen- 
sions for their lengths and position \\ith respect to each other. 

51. Develop the surfaces of Question 50. 

52. Two cyhnders of 2" diameter and \\" diameter intersect at an 

angle of 60°. Find the line of intersection of the surfaces. As- 
sume suitable dimensions for their lengths and position with 
respect to each other. 

53. Develop the surfaces of Question 52. 

54. Two cylinders of 2" diameter and 1 \" diameter intersect at an angle of 

45°. Find the line of intersection of the surfaces. Assume suitable 
dimensions for their lengths and position with respect to each other. 

55. Develop the surfaces of Question 54. 

56. Two cylinders of 2" diameter and I4" diameter intersect at an angle 

of 30°. Find the line of intersection of the surfaces. Assume 
suitable dimensions for their lengths and position with respect 
to each other. 

57. Develop the surfaces of Question 56. 

58. Two cylinders of 2" diameter intersect at a right angle and have 

their axes offset \" . Find the line of intersection of tlie surfaces. 
Assume the additional dimensions. 

59. Develop the surface of one of tlie cylinders of Question 58. 

60. A 2" cylinder intersects a \\" cylinder so tliat their a.xes art^ offset 

i" and make a right angle with each otlier. Find the line of 
intersection of the surfaces. Assume tlie additional dimensions. 



230 



GEOMETRICAL PROBLEMS IN PROJECTION 



I". The elements intersect 
intersection of the surfaces. 



61. Develop the surfaces of Question 60. 

62. Two 2" cyhnders have their axes offset 

at an angle of 60°. Find the line of 
Assume the additional dimensions. 

63. Develop the surfaces of Question 62. 

64. Two 2" cyhnders have their axes offset 

at an angle of 45°. Find the Une of intersection of the surfaces. 
Assume the additional dimensions. 

65. Develop the surfaces of Question 64. 

66. Two 2" cyhnders have their axes offset 

at an angle of 30°. Find the line of 
Assume the additional dimensions. 

67. Develop the surfaces of Question 66. 

68. A 2" cylinder intersects a If" cyhnder 

axes are offset \" . Find the hne of 
Assume the additional dimensions. 

69. Develop the surfaces of Question 68. 



\" . The elements intersect 
intersection of the surfaces. 



at an angle of 60°. Their 
intersection of the surfaces. 





Fig. 12-B. 



Fig. 12-C. 



70. A 2" cylinder intersects a If" cyhnder at an angle of 45°. Their 
axes are offset J". Find the line of intersection of the surfaces. 
Assume the additional dimensions. 

7L Develop the surfaces of Question 70. 

72. A 2" cylinder intersects a If" cyhnder at an angle of 30°. Their 

axes are offset i". Find the hne of intersection of the surfaces. 
Assume the additional dimensions. 

73. Develop the surfaces of Question 72. 

74. Find the hne of intersection of the surfaces of the two cones shown 

in Fig. 12-B. Assume suitable dimensions. 

75. Develop the surfaces of Question 74. 

76. Assume a pair of cones similar to those shown in Fig. 12-B, but, 

with equal altitudes. Then, find the hne of intersection of the 
surfaces. 

77. Develop the surfaces of Question 76. 

78. Assume a cone and cyhnder similar to that shown in Fig. 12-C. 

Then, find the hne of intersection of the surfaces. 

79. Develop the surfaces of Question 78. 



INTERSECTIONS OF SURFACES WITH EACH OTHER 231 

80. Assume a cone and cylinder similar to that shown in Fig. 12-C, 

but, have the cylinder elliptical. Then, find the fine of inter- 
section of the surfaces. 

81. Develop the surfaces of Question 80. 

82. Assume a cone and cylinder similar to that shown in Fig. 12-C, 

but, have the cone elliptical. Then, find the line of intersection 
of the surfaces. 

83. Develop the surfaces of Question 82. 

84. Assume an arrangement of cone and cyhnder somewhat similar 

to that of Fig. 12-C, but, have both cone and cylinder elhptical. 
Then, find the fine of intersection of the surfaces. 

85. Develop the surfaces of Question 84. 

86. Assume a cyhnder and cone of revolution, whose general direction 

of axes are at right angles, but, which are offset as shown in 
Fig. 12-D. Then, find the fine of intersection of the sur- 
faces. 

87. Develop the surfaces in Question 86, 





Fig. 12-D. 



Fig. 12-E. 



88. Assume an elliptical cylinder and a cone of revolution, whose general 

direction of axes are at right angles to each other, but, which 
are offset as shown in Fig. 12-D. Then, find the fine of inter- 
section of the surfaces. 

89. Develop the surfaces of Question 88. 

90. Assume a circular cyhnder of revolution and an elliptical cone, whose 

general direction of axes are at right angles to each other, but, 
which are offset as shown in Fig. 12-D. Then, find the fine of 
intersection of the surfaces. 

91. Develop the surfaces of Question 90. 

92. Assume an elliptical cylinder and an elhptical cone, whose general 

direction of axes, are at right angles to each other, but, which 
are offset, as shown in Fig. 12-D. Then, find the line of inter- 
section of the surfaces. 

93. Develop the surfaces of Question 92. 

94. Assume a cylinder and cone of revolution, similar to that of Fig. 

12-E, and make angle a =60°. 
of the surfaces. 

95. Develop the surfaces of Question 94. 



Then, find the lino of intersection 



232 



(GEOMETRICAL PROBLEMS IN PROJECTION 



96. Assume a cl3ander and cone of revolution, similar to that of Fig. 

12-E, and make angle a =45°. Then, find the Hne of inter- 
section of the surfaces. 

97. Develop the surfaces of Question 96. 

98. Assume a cyhnder and a cone of revolution, similar to that of Fig. 

12-E, and make angle a =30°. Then, find the line of intersection 
of the surfaces. 

99. Develop the surfaces of Question 98. 

100. Assume an elhptical cyhnder and a cone of revolution arranged 

similar to that of Fig. 12-E, and make the angle a =60°. Then, 
. find the hne of intersection of the surfaces. 

101. Develop the surfaces of Question 100. 

102. Assume a cylinder of revolution and an elliptical cone, arranged 

similar to that of Fig. 12-E, and make the angle a =45°. Then, 
find the line of intersection of the surfaces. 

103. Develop the surfaces of Question 102. 






Fig. 12-E. 



Fig. 12-F. 



Fig. 12-G. 



104. Assume an elhptical cyhnder and an elliptical cone arranged similar 

to that of Fig. 12-E, and make the angle a =60°. Then find 
the line of intersection of the surfaces. 

105. Develop the surfaces of Question 104. 

106. Assume a cyhnder and cone of revolution as shown in Fig. 12-F, 

and make angle a =60°. Then, find the hne of intersection of 
the surfaces. 

107. Develop the surfaces of Question 106. 

108. Assume a cyhnder and a cone of revolution as shown in Fig. 12-F, 

and make angle a =45°. Then, find the hne of intersection of 
the surfaces. 

109. Develop the surfaces of Question 108. 

110. Assume a cylinder and a cone of revolution as shown in Fig. 12-F, 

and make angle a = 30°. Then, find the line of intersection of 
the surfaces. 

111. Develop the surfaces of Question 110. 

112. Assume an elhptical cyhnder and a cone of revolution, arranged 

similar to that shown in Fig. 12-F, and make angle a =60°. 
Then find the hne of intersection of the surfaces. 



INTERSECTIONS OF SURFACES WITH EACH OTHER 233 

113. Develop the surfaces of Question 112. 

114. Assume a cylinder of revolution and an elliptical cone, arranged 

similar to that shown in Fig. 12-F, and make angle a =45°. 
Then, find the line of intersection of the surfaces. 





Fig. 12-H. 



Fig. 12-I. 



115. Develop the surfaces of Question 114. 

116. Assume an elliptical cyhnder and an elliptical cone, somewhat 

similar to that shown in Fig, 12-F, and make angle a =60°. 
Then find the fine of intersection of the surfaces. 

117. Develop the surfaces of Question 116. 

118. Assume a cone and cyhnder as shown in Fig. 12-G. Then, find 

the line of intersection of the surfaces. 

119. Develop the surfaces of Question 118. 

120. Assume a cyhnder and a sphere as shown in Fig. 12-H. Then, 

find the line of intersection of the sur- 
faces. 

121. Develop the surface of the cylinder of Ques- 

tion 120. 

122. Develop the surface of the cyhnder of Ques- 

tion 120, and, also, the surface of the 
sphere by the gore method. 

123. Assume a cylinder and sphere as shown in 

Fig. 12-1. 'Then, find the line of intersec- 
tion of the surfaces. 

124. Develop the surface of the cylinder in Ques- 

tion 123. Fio. 12-J. 

125. Develop the surface of tlic cylinder in (Ques- 

tion 123, and, also, the surface of the sphere by the zone method. 

126. Assume a cone and a sphere as shown in Fig. 12-J. Then, find 

the line of intersection of the surfaces. 

127. Develop the surface of the cone in (Question 126. 

128. Develop the surface of the cone in Question 126, and, also, the 

sphere by the gore method. 





234 



GEOMETRICAL PROBLEMS IN PROJECTION 



129. Assume two doubly curved surfaces of revolution, whose axes 

intersect. Then, find the line of intersection of the surfaces. 

130. Develop the surfaces of Question 129 by the gore method. 

131. Develop the surfaces of Question 129 by the zone method. 

132. Assume two doubly curved surfaces of revolution whose axes do 

not intersect. Then, find the line of intersection of the surfaces. 





/ 

ua 



Fig. 12-K. 



133. Develop the surfaces of Question 132 by the gore method. 

134. Develop the surfaces of Question 132 by the zone method. 

135. Assume a frustum of a cone as shown in Fig. 12-K. A circular 

cjdinder is to be fitted to it, as shown. Develop the surfaces. 
Hint. — Find vertex of cone before proceeding with the devel- 
opment. 



PART III 

PRINCIPLES OF CONVERGENT PROJECTING-LINE 

DRAWING 



CHAPTER XIII 
PERSPECTIVE PROJECTION 

1301. Introductory. Observation shows that the apparent 
magnitude of objects is some function of the distance between 
the observer and the object. The drawings made according to 
the principles of parallel projecting-line drawing, and treated in 
Part I of this book, make no allowance for the observer's posi- 
tion with respect to the object. In other words, the location of 
the object with respect to the plane of pro jecj^ion has no influence 
on the size of the resultant picture, provided that the inclina- 
tion of the various lines on the object, to the plane of projec- 
tion, remain the same. Hence, as the remoteness of the object 
influences its apparent size, then, as a consequence, parallel 
projecting-line drawings must have an unnatural appearance. 
The strained appearance of drawings of this type is quite notice- 
able in orthographic projection wherein two or more views must 
be interpreted simultaneously, and is less noticeal)le in the case 
of oblique or axonometric projection. On the other hand, the 
rapidity with which drawings of the parallel projecting-line typo 
can be made, and their adaptability for construction purposes, 
are strong points in their favor. 

1302. Scenographic projection. To overcome tlu foregoing 
objections, and to present a drawing to the readcT which cor- 
rects for distance, scenographic projections are used. To make 
these, convergent projecting lines are used, and tlu* obstTvor 
is located at the point of convergency. The surface on wliich 

235 



236 CONVERGENT PROJECTING-LINE DRAWING 

iscenographic projections are made may be spherical, cylindrical, 
etc., and find their most extensive application in decorative 
painting. 

1303. Linear perspective. In the decoration of an interior 

of a dome or of a cylindrical wall, the resulting picture should 
be of such order as to give a correct image for the assumed loca- 
tion of the eye. In engineering drawing, there is little or no use 
for projections on spherical, cylindrical, or other curved surfaces. 
When scenographic projections are made on a plane, this type of 
projection is called linear perspective. A linear perspective, 
therefore, may be defined as the drawing made on a plane surface 
by the aid of convergent projecting lines, the point of convergency 
being at a finite distance from the object and from the plane. 
The plane of projection is called the picture plane, and the posi- 
tion of the eye is referred to as the point of sight. 

1304. Visual rays and visual angle. If an object — an arrow 
for instance — be placed a certain distance from the eye, say in 

the position ab (Fig. 227), and c be 
the point of sight, the two extreme 
rays of light or visual rays, ac and 
be, form an angle which is knoT\Ti as 
the visual angle. If, now, the arrow be 
Fig. 227. moved to a more remote position de, 

the limiting visual rays dc and ec 
give a smaller visual angle. The physiological effect of this 
variation in the visual angle is to alter the apparent size of the 
object. If one eye is closed during this experiment, the distance 
from the eye to the object cannot be easily estimated, but the 
apparent magnitude of the object will be some function of the 
visual angle. In binocular vision, the muscular effort required 
to focus both eyes on the object will, wdth some experience, enable 
an estimate of the distance from the eye to the object; the mind 
will automatically correct for the smaller visual angle and greater 
distance, and, thus, to an experienced observer, give a more or 
less correct impression of the actual size of the object. This 
experience is generally limited to horizontal distances only, as 
very few can estimate correctly the diameter of a clock on a church 
steeple unless they have been accustomed to making such observa- 
tions. As the use of two eyes in depicting objects in space causes 




PEPtSPECTIVE PROJECTION 237 

slightly different inpressions on different observers, the eye 
will be assumed hereafter as a single point. 

1305. Vanishing point. When looking along a stretch of 
straight railroad tracks, it may be noted that the tracks appar- 
ently vanish in the distance. Likewise, in viewing a street of 
houses of about the same height, the roof line appears to meet 
the sidewalk line in the distance. It is known that the actual 
distances between the rails, and between the sidewalk and roof 
is always the same; yet the visual angle being less in the distant 
observation, gives the impression of a vanishing point, which 
may, therefore, be defined as the point where parallel lines seem 
to vanish. At an infinite distance the visual angle is zero, and, 
hence, the parallel lines appear to meet in a point. The lines 
themselves do not vanish, but their perspective projections 
vanish. 

1306. Theory of perspective projection. The simplest 
notion of perspective drawing can be obtained by looking at a 
distant house through a window-pane. If the observer would 
trace on the window-pane exactly what he sees, and locate all 
points on the pane so that corresponding points on the house are 
directly behind them, a true linear perspective of the house would 
be the result. Manifestly, the location of each successive point 
is the same as locating the piercing point of the visual ray on 
the picture plane, the picture plane in our case being the window- 
pane. 

1307. Aerial perspective. If this perspective were now 
colored to resemble the house beyond, proper attention being 
paid to light, shade and shadow, an aerial perspective would be 
the result. In brief, this aerial perspective would present to the 
eye a picture that represents the natural condition as near as the 
skill of the artist will permit. In our work the Hnear perspective 
will alone be considered, and will be denoted as the *' persj)ective," 
aerial perspective finding little application in engineering. 

1308. Location of picture plane. It is customary in per- 
spective to assume that the picture ])lan(^ is situated Ix^tween the 
eye tyid the ol)ject. Under these* conditions, the picture is smaller 
than the object, and usually this is necessary. Th(> eve is assumed 

to he in the first angle; the* ()))j(M't. howcviM". is geiKM'nlly in the 



238 



CONVERGENT PEOJECTING-LINE DRAWING 



second angle * for reasons noted above. The vertical plane is 
thus the picture plane. A little reflection will show that this is 
in accord with our daily experience. Observers stand on the 
ground and look at some distant object; visual rays enter the 
eye at various angles; the mean of these rays is horizontal or 
nearly so, and, therefore, the projections naturally fall on a 
vertical plane interposed in the line of sight. The wdndow- 
pane picture alluded to is an example. 

1309. Perspective of a line. Let, in Fig. 228, AB be an 
arrow, standing vertically as shown in the second angle; the 
point of sight C is located in the first angle. The visual rays 
pierce the picture plane at a." and b", and a"b", in the picture 
plane, is the perspective of AB in space. The similar condition 




Fig. 228. 



Fig. 229. 



of Fig. 228 in orthographic projection is represented in Fig. 
229. The arrow and the point of sight are showm by their pro- 
jections on the horizontal and vertical planes. The arrow being 
perpendicular to the horizontal plane, both projections fall at 
the same point ab; the vertical projection is shown as a'b'; the 
point of sight is represented on the horizontal and vertical 
planes as c and c', respectively. Any visual ray can likeT^'ise be 
represented by its projections, and, therefore, the horizontal 
projection of the point of sight is joined w^th the horizontal pro- 

* The object may be located in any angle; the principles are the same 
for all angles. It is perhaps more convenient to use the second angle, as 
the hnes do not have to be extended to obtain the piercing points as would 
be the case for first-angle projections. 



PERSPECTIVE PROJECTION 



239 



jections. Similarl}^, the same is true of the vertical projections 
of the arrow and point of sight. 

It is now necessary to find the piercing points (506,805) of 
the visual rays on the picture plane (vertical plane); and these 
are seen at a" and b". Therefore, a"b" is the perspective of 
AB in space. 

1310. Perspectives of lines perpendicular to the hori= 
zontal plane. In Fig. 230, several arrows are shown, in projec- 
tion, all of which are perpendicular to the horizontal plane. Their 
perspectives are a"b'', d"e'^ g"h'\ On observation, it will 
be noted that all the perspectives of lines perpendicular to the 
horizontal plane are vertical. This is true, since, when a plane 





Fig. 230. 



Fig. 231. 



is passed through any line and revolved until it contains the 
point of sight, the visual plane (as this plane is then called) is 
manifestly perpendicular to the horizontal plane, and its vertical 
trace (603) is perpendicular to the ground line. 

1311. Perspectives of lines parallel to both principal 
planes. Fig. 231 shows two arrows parallel to both planes, the 
projections and perspectives being designated as before. This 
case shows that all perspectives are parallel. To prove this, 
pass a plane through the line in space and the point of sight; 
the vertical trace of the plane so obatined will be parallel to 
the ground line * because the line is parallel to the groiuul 
line. 



* If (ho lino in space lies in the ground line, the vcMticai tiac(> will ooincido 
with tho f:;round lino. Tliis is cvidciilly ;i .s|)(>ci;d ca.so. 



240 



CONVERGENT PROJECTING-LINE DRAWING 



1312. Perspectives of lines perpendicular to the picture 
plane. Suppose a series of lines perpendicular to the vertical 
plane is taken. This case is illustrated in Fig. 232. The per- 
spectives are drawn as before and designated as is customary. 
It may now be observed that the perspectives of all these lines 




Fig. 232. 



(or arrows) vanish in the vertical projection of the point of sight. 
This vanishing point for the perspectives of all perpendiculars 
to the picture plane is called the centre of the picture. The 
reason for this is as follows: It is known that the visual angle 




Fig. 233. 



Fig. 234. 



becomes less as the distance from the point of sight increases; 
at an infinite distance the angle is zero, the perspectives of the 
lines converge, and, therefore, in our nomenclature vanish. 

1313. Perspectives of parallel lines, inclined to the 
picture plane. Assume further another set of lines, Fig. 233, 



PERSPECTIVE PROJECTION 241 

parallel to each other, inclined to the vertical plane, but parallel 
to the horizontal plane. Their orthographic projections are 
parallel and their perspectives will be seen to converge to the 
point V. Like the lines in the preceding paragraph, they vanish 
at a point which is the vanishing point of any number of lines 
parallel only to those assumed. The truth of this is established 
by the fact that the visual angle becomes zero (1304) at an infinite 
distance from the point of sight and, hence, the perspectives 
vanish as shown. To determine this vanishing point, draw 
any line through the point of sight parallel to the system 
of parallel lines; and its piercing point on the vertical or 
picture plane will be the required vanishing point. It 
amounts to the same thing to say that the vanishing point 
is the perspective of any line of this system at an infinite dis- 
tance. 

A slightly different condition is shown in Fig. 234. The 
lines are still parallel to each other and to the horizontal plane, 
but are inclined to the vertical plane in a different direction. 
Again, the perspectives of these lines will converge to a new van- 
ishing point, situated, however, much the same as the one just 
preceding. 

1314. Horizon. On observation of the cases cited in Art. 
1313, it can be seen that the vanishing point lies on a horizontal 
line through the vertical projection of the point of sight. This 
is so since a line through the point of sight parallel to either sys- 
tem will pierce the picture plane in a point somewhere on its 
vertical projection. As the line is parallel to the horizontal 
plane, its vertical projection will be parallel to the ground line 
and, as shown, will contain all vanishing points of all systems of 
horizontal lines. Any one system of parallel lines will have 
but one vanishing point, but as the lines may slope in various 
directions, and still be parallel to the horizontal plane, every 
system will have its own vanishing ])oiut and the horizontal Hnt> 
drawn through the vertical projection of the point of sight will 
be the locus of all these vanishing points. This horizontal 
line through the vertical projection of the point of sight is ('nlUnl 
the horizon. 

As a corollary to \\\v above, it may be stated, that all plaiu^s 
])arallel to the horizontal i)lane vanish in the horizon. As l>(>fort\ 



242 



CONVERGENT PROJECTING-LINE DRAWING 



the visual angle becomes less as the distance increases, and, hence, 
becomes zero at infinity. 

There are an unlimited number of systems of lines resulting 
in an unlimited number of vanishing points, whether they are 
on the horizon or not. In most drawings the vertical and hori- 
zontal Hues are usually the more common. The lines perpendicular 
to the picture plane are horizontal lines, and, therefore, the centre 
of the picture (vanishing point for the perspectives of the per- 
pendiculars) must be on the horizon. 

1315. Perspective of a point. The process of finding the 
perspective of a line of definite length is to find the perspectives 





Fig. 235. 



Fig. 236. 



of the two extremities of the line. When the perspectives of the 
extremities of the line are found, the line joining them is the 
perspective of the given line. In Fig. 235, two points A and B 
are chosen whose perspectives will be found to be a." and b''. 

When the point lies in picture plane it is its own perspective. 
This is shown in Fig. 236, where a! is the vertical projection and 
a is therefore in the ground line; hence, the perspective is a'. 

In the construction of any perspective, the method is to 
locate the perspectives of certain points which are joined by their 
proper lines. The correct grouping of lines determine certain 
surfaces which, when closed, determine the required soUd. 

1316. Indefinite perspective of a line. Let AB, Fig. 237, 
be a limited portion of a line FE. Hence, by obtaining the 



PEKSPECTIVE PROJECTION 243 

piercing points of the visual rays AC and BC, the perspective 
a''b" is determined. Likewise, DE is another limited portion 
of the line FE and its perspective is &"e". If a line be drawn 
through the point of sight C, parallel to the line FE, then, its 
piercing point g' on the picture plane will be the vanishing point 
of the perspectives of a system of lines parallel to FE (1313). 
Also, as FE pierces the picture plane at f , then i' will be 
its own perspective (1315). Therefore, if a line be made to 
join t' and g', it will be the perspective of a line that reaches 
from the picture plane out to an infinite distance. And, also, 
the perspective of any limited portion of this line must lie 




Fig. 237. 

on the perspective of the line. From the construction in 
Fig. 237, it will be observed that a"b" and d"e" lie on the 
line f'g'. 

Consequently, the line f g' is the indefinite perspective of a 
line FE which reaches from the picture plane at t' out to an 
infinite distance. Thus, the indefinite perspective of a line may be 
defined as the perspective of a line that reaches from the picture 
plane to infinity. 

Before leaving Fig. 237, it is desirable to note that g' is the 
vanishing point of a system of lines parallel to FE. It is not 
located on the horizon because the line FE is not parallel to the 
horizontal plane. 



244 



CONVERGENT PROJECTING-LINE DRAWING 



1317. Problem 1. To find the perspective of a cube by means 
of the piercing points of the visual rays on the picture plane. 

Construction. Let ABCDEFGH, Fig. 238, be the eight 
corners of the cube, and let S be the point of sight. The cube is 
in the second angle and therefore both projections are above 
the ground line (315, 317, 514, 516). The horizontal projec- 
tion of the cube is indicated by abed for the upper side, and efgh 
for the base. The vertical projections are shown as a'b'c'd' 
for the upper side, and e'f'g'h' for the base. The cube is located 
in the second angle so that the two projections overlap. The 
point of sight is in the first angle and is showni by its horizontal 
projection s and its vertical projection s'. Join s with e and 




Fig. 238. 



obtain the horizontal projection of the visual ray SE in space; 
do likewise for the vertical projection with the result that s'e' 
will be the visual ray projected on the picture plane. The point 
where this pierces the picture plane is E, and this is one point 
of the required perspective. Consider next point C. This 
is found in a manner similar to point E just determined. The 
horizontal projection of the visual ray SC is so and the vertical 
projection of the visual ray is s'c' and this pierces the vertical 
or picture plane in the point C as shown. In the foregoing man- 
ner, all other points are determined. By joining the correct 
points with each other, a linear perspective will be obtained. 

It will be noticed that CG, BF, DH, and AE are vertical 
because the lines in space are vertical (1310), and hence their 
perspectives are vertical. This latter fact acts as a check after 



PERSPECTIVE PROJECTION 245 

locating the perspectives of the upper face and the base of the 
cube. It will be noticed, further, that the perspectives of the 
horizontal lines in space meet at the vanishing points V and V 
(1313, 1314). 

1318. Perspectives of intersecting lines. Instead of locat- 
ing the piercing point of a visual ray by drawing the projections 
of this ray, it is sometimes found desirable to use another method. 
For this purpose another principle must be developed. 

If two lines in space inter- ^z 
sect, their perspectives inter- 
sect, because the perspective 
of a line can be considered as 
being made up of the perspec- 
tives of all the points on the 
line. As the intersection is a 
point common to the two 
lines, a visual ray to this point 
should pierce the picture plane 
in the intersection of the 
perspectives. Reference to 
Fig. 239 will show that such jtjq 239. 

is the case. CG is the visual 

ray in space of the point of intersection of the two lines, and its 
perspective is g", which is the intersection of the perspectives 
a'V and d''e''. 

1319. Perpendicular and diagonal. Obviously, if it is desired 
to find the perspective of a point in space, two linos can be drawn 
through the point and the intersection of their perspectives found. 
The advantage of this will appear later. The two Unes generally 
used are: first, a perpendicular, which is a hne perpendicular 
to the picture plane and whose perspective therefore vanishes 
in the vertical projection of the point of sight (1312) ; and second, 
a diagonal which is a horizontal line making an angle of 45° with 
the picture plane, and whose perspective vanishes somewhere 
on the horizon (1313). As the perpendicular and diagonal 
drawn through a point are both parallc^l to the horizontal plane, 
these two intersecting lines determine a plane which, like all 
other horizontal planes, vanish in the horizon. Instead of using 
a diagonal making 45°, any otluT angle may be used, provided 




246 COXVERGEXT PROJECTIXG-LINE DRAWING 

it is less than 90^. This latter line would be parallel to the picture 
plane and therefore would not pierce it. 

It can also be observed that there are two possible diagonals 
through any point, one whose perspective vanishes to the left 
of the point of sight and the other whose persj)ective vanishes 
to the right of the point of sight. 

It is further kno-^Ti that the perspectives of all parallel lines 
vanish in one point and therefore the perspectives of all parallel 
diagonals through any point must have a common vanishing 
point. With two possible diagonals through a given point in 
space, two vanishing points are obtained on the horizon. 

1320. To find the perspective of a point b\ the method 
of perpendiculars and diagonals. Let a and a', Fig. 240. 

Horizon v 



a 

m a 




Fig. 240. 

represent the projections of a point in the second angle, and let c 
and c' be the projections of the point of sight. By drawing the 
\'isual rays ac and a'c' the piercing point a" is determined by 
erecting a perpendicular at e as shown. Although the perspective 
a" is determined by this method, it may also be determined by 
finding the intersections of the perspectives of a perpendicular 
and a diagonal through the point A in space. 

Thus, by drawing a line ab, making a 45^ angle with the 
groimd line, the horizontal projection of the diagonal is found. 
As the diagonal is a horizontal line, its vertical projection is 
a'b'. This diagonal pierces the pictiu-e plane at b'. which point 
is its 0"v\TL perspective. Its perspective also vanishes at V, the 
vanishing point of all horizontal lines whose inclination to the 
picture plane is at the 45° angle shown and whose directions are 
parallel to each other. The poiut V is foimd by drawing a line 



PERSPECTIVE PROJECTION 



247 



cv parallel to ab, c'V parallel to a'b', and then finding the piercing 
point V. The horizontal projection of a perpendicular through 
A is ad; its vertical projection is evidently a', which is also 
its piercing point on the picture plane and as it lies in the picture 
plane, it is hence its own perspective. The vanishing point of the 
perspective of the perpendicular is at c', the centre of the picture. 
Since the perspective of a given point must lie on the per- 
spectives of any two lines drawn through the point, then, as 
b'V is the indefinite perspective of a diagonal, and a'c' is the 
indefinite perspective of a perpendicular, their intersection a'' is 
the required perspective of the point A. The fact that a" has 
already been determined by drawing the visual ray and its 



V Horizon 




Fig. 241. 



piercing point found shows that the construction is correct, 
and that either method will give the same result. 

In Fig. 241, a similar construction is shown. The points 
V and V are the vanishing points of the left and right diagonals 
respectively. The perspective a'' may be determined by the 
use of the two diagonals without the aid of the perpendicular. 
For instance ab and a'b' are corresponding projections of one 
diagonal; ad and a'd' are corresponding projections of the other 
diagonal. Hence b'V is the indefinite perspective of the right 
diagonal (since cv' is drawn to the right) and d'V is the indefinite 
perspective of the left diagonal. Their intersection determines 
a'', the required perspective of the point A. The indefinite \)vr- 
spective of the per])(^n(H('ular is shown as a'c' and i)asses 
through the point a" as it sliould. 



248 



CONVERGENT PROJECTING-LINE DRAWING 



Any two lines may be used to determine the perspective and 
should be chosen so as to intersect as at nearly a right angle as 
possible to insure accuracy of the location of the point. The 
visual ray may also be drawn on this diagram and its piercing 
point will again determine ^." . 




Fig. 242. 



1321. To find the perspective of a line by the method 
of perpendiculars and diagonals. Suppose it is desired to 
construct the perspective of an arrow by the method of perpendic- 
ulars and diagonals. Fig. 242 shows a case of this kind. AB 

is an arrow situated in the second 
angle; C is the point of sight. 
A perpendicular through the 
point of sight pierces the picture 
plane in the vertical projection c'. 
The two possible diagonals whose 
horizontal projections are given 
by cm and en pierce the picture 
plane in V^ and V respectively. 
The perspective of any diagonal 
drawn through any point in 
space will vanish in either of these vanishing points depending 
upon whether the diagonal is drawn to the right or to the left 
of the point in question. Likemse, the perspective of a per- 
pendicular drawn through any point in space will vanish in the 
centre of the picture. 

In the case at issue, the horizontal projection of a diagonal 
through b is bo and its vertical projection is b'o'; the piercing 
points is at o'. As the perspectives of all lines parallel to the 
diagonal have a common vanishing point, then the perspective 
of the diagonal through B must vanish at V and the perspective 
of the point B must be somewhere on the hne joining o' and V. 
Turning attention to the perpendicular through B, it is found 
that its vertical projection corresponds wdth the vertical pro- 
jection of the point itself and is therefore b'. The perspectives 
of all perpendiculars to the vertical or picture plane vanish in 
the centre of the picture; and, hence, the perspective of B in 
space must be somewhere on the line b'c'. It must also be 
somewhere on the perspective of the diagonal and, hence, it is 
at their intersection b''. This can also be shown by drawing 



PERSPECTIVE PROJECTION 



249 



the visual ray through the point of sight C and the point B. 
The horizontal and vertical projections of the visual ray are 
cb and c'b' and they pierce the picture plane at b", which is the 
same point obtained by finding the indefinite perspectives of 
the perpendicular and the diagonal. 

By similar reasoning the perspective a'' is obtained and, 
therefore, a'^b'' is the perspective of AB. 

1322. Revolution of the horizontal plane. The fact that 
the second and fourth angles are not used in drawing on account 
of the conflict between the separate views has already been con- 
sidered (315, 317). This is again shown in Fig. 238. The two 
views of the cube overlap and make deciphering more difficult. 
To overcome this diflaculty the horizontal plane is revolved 




Fig. 243. 



180° from its present position. This brings the horizontal 
projections below the ground line and leaves the vertical pro- 
jections the same as before. Now the diagonals through any 
point slope in the reverse direction, and care must therefore be 
used in selecting the proper vanishing point while drawing the 
indefinite perspective of the diagonal. 

1323. To find the perspective of a point when the hori- 
zontal plane is revolved. Let Fig. 243 represent the conditions 
of the problem. The vertical projection of the given point is 
a' and its corresponding horizontal j^rojection is at a, below the 
ground line due to the 180° revolution of the horizontal \)\i\m\ 
The vertical projection of the point of sight is at c' and its cor- 
responding horizontal projection is at c, now above the ground 
line. The conditions are sucli that the given point seems like 
a first angle projection and the point of sight seems like a second 



250 



CONVERGENT PROJECTING-LINE DRAWING 



angle projection, whereas the actual conditions are just the reverse 
of this. 

Through a, draw ab the horizontal projection of the diagonal; 
the corresponding vertical projection is a'W with b' as the piercing 
point. As ab is drawn to the right, the indefinite perspective 
of the diagonal must vanish in the left vanishing point at V, 
Hence, draw Vb', the indefinite perspective of the diagonal. 




Fig. 243. 



The indefinite perspective of the perpendicular remains unchanged 
and is shown in the diagram as a'c'. Therefore the intersection 
of Vb' and a'c' determine a'', the required perspective of the point 
A. The accuracy of the construction is checked by drawing the 
visual ray ca and then erecting a perpendicular at o, as shown, 
which passes through a'' as it should. 

1324. To find the perspective of a line when the hori= 
zontal plane is revolved. The point B in space will again be 

located in Fig. 244 just as was 
done in Fig. 242. The horizontal 
projection of the diagonal is bo 
and its vertical projection is b'o' ; 
the piercing point, therefore, is 
o'. The horizontal projection of 
the point of sight c is now above 
the ground line instead of below 
it, due to the 180° revolution of 
■p 244 ^^^ horizontal plane; the orig- 

inal position of this point is 
marked c'\ A diagonal through the point of sight, parallel to 




PERSPECTIVE PROJECTION 251 

the diagonal through the point B, is shown as en and pierces 
the vertical plane at V and, therefore, o'V is the indefinite per- 
spective of the diagonal. The indefinite perspective of the 
perpendicular, as before, is b'c', vanishing in the centre of the 
picture. The intersection of these two indefinite perspectives 
determines b'', the perspective of the point B in space. 

Instead of using the left diagonal bo, through b, it is possible 
to use the right diagonal bq, and this pierces the picture plane 
at q', and vanishes at V which (as must always be the case) 
again determines the perspective b". It must be observed that 
it matters little which diagonal is used with a perpendicular, 
or, whether only the diagonals without the perpendicular are 
used. In practice such lines are selected as will intersect as 
nearly as possible at right angles since the point of intersec- 
tion is thereby more accurately determined than if the two lines 
intersected acutely. Whatever is done the principal points 
can always be located by any two lines, and the other may be used 
as a check. Still another check can be had by drawing the hori- 
zontal projection of the visual ray cb whereby b'', the perspective, 
is once more determined. The point A in space is located in an 
identical manner. 

1325. Location of diagonal vanishing points. On observa- 
tion of Figs. 242 and 244, it is seen that the distance V to c' and 
V to c' is the same as the distance of the point of sight is from 
the vertical plane. This is true because a 45° diagonal is used 
and these distances are the equal sides of a triangle so formed. 
The use of any other angle would not give the same result, although 
the distance of the vanishing points from the centre of the pic- 
ture would always be equal. 

All the principles that are necessary for the drawing of any kind 
of a linear perspective have now been developed. The subsequent 
problems will illustrate their uses in a variety of cases. Certain 
adaptations required for commercial application will apixvir 
subsequently. 

1326. Problem 2. To find the perspective of a cube by the 
method of perpendiculars and diagonals. 

Construction. L(>i ABCDEFGH in Fig. 245 represent the 
cube. A case has \)vvn s(>le('te(l that is identical to the one 
shown in Fig. 238, in order that the dilfereii('(> btMween the two 



252 



CONVERGENT PROJECTING-LINE DRAWING 



methods may be clearly illustrated. Suppose it is desired to 
locate the perspective of the point E. Draw from e, in the hori- 
zontal projection, the diagonal eo, the vertical projection is e'o' 
with o' as a piercing point. Join o' with v, and the indefinite 
perspective of the diagonal is obtained. The perpendicular 
from e pierces the vertical plane at e' and e's' is the indefinite 
perspective. The intersection E of these indefinite perspectives 
is the required perspective of the point. This point can also be 
checked by drawing the other diagonal which pierces the picture 
plane at p', thereby making pV the indefinite perspective of 
the latter diagonal, which passes through E, as it should. 

The point C is located by drawing the perpendicular and 




Fig. 245. 



diagonal in the same way as was done for the point E. The 
rest of the construction has been omitted for the sake of clearness. 
As in Fig. 242, the vanishing points V and V are laid out by 
drawing a line through the point of sight parallel to the system 
of lines. Fig. 245 does not show the construction in this way 
because the horizontal projection of the point of sight is now 
above the ground line, due to the 1 80 °-re volution of the horizon- 
tal plane. Instead, the fines have been draA\Ti from the original 
horizontal projection shown as s" and care has been taken to sea 
that the lines are parafiel to the reversed position of the horizon- 
tal projection. A reference to both Figs. 238 and 245 will make 
this all clear. 

In Fig. 245 two vanishing points v' and v will be noted which 



PERSPECTIVE PROJECTION 



253 



are (as has already been shown to be) the vanishing points of the 
perspectives of all diagonals. It is also known that the centre 
of the picture s' is the vanishing point of the perspectives of all 
perpendiculars. Altogether, there are five vanishing points; 
the points V and V are the vanishing points of the perspectives 
of the horizontal lines on the cube. 

1327. Problems. To find the perspective of a hexagonal prism. 

Construction. Reference to Fig. 246 shows that one edge 
of the prism hes in the picture plane while the base is in the hor- 
izontal plane.* The diagonals used to determine the vanishing 
points are here chosen to be 30°, as in this way they are also 
parallel to some of the lines of the object and, therefore, have 




common vanishing points. V and V are the vanishing points 
of the diagonals, and of those sides which make an angle of 30° 
with the picture plane. The centre of the picture is at s' and, 
therefore, the vanishing point of the perspectives of all perpendic- 
ulars. The edge AG lies in the picture plane, and, therefor(\ 
is shown in its actual size. The indefinite perspectives of th(^ 
diagonals from the extremities A and G will give tlie dircH'tiou 
of the sides FA and AB for the top face and LG and GH for iUv 
base. The points F and L are located by drawing a diagonal 
and perpendicular through f and 1 in the horizontal i)roj(H'ti()n. 
The distance mm' and nn' is equal to AG, the heiglit of the ])risin. 

* This |)i(;(ui(> lulls ;i .slniiiicd uppoariinco duo to the selection of (he point 
of sifrht. See Art. 1331 in this coiHu^et ion. 



254 



CONVERGENT PROJECTING-LINE DRAWING 



1328. Problem 4. To find the perspective of a pyramid 
superimposed on a square base. 

Construction. In Fig. 247, the edge GK is shown in the 
picture plane, a fact which enables the immediate determination 
of the direction of JK, KL, FG and GH by joining G and K with 
the vanishing points V and V. It is only necessary to show the 
points J and L to determine the verticals FJ and HL. The 
method for this has already been shown. One other important 
point to locate is the apex A. This is shown by drawing a diagonal 
and perpendicular through a. The line AO, in space, pierces the 




Fig. 247. 

picture plane at o', and AP at p' at a height oo' equal to the height 
of the apex above the horizontal plane. Joining p' with s\the 
indefinite perspective of the perpendicular is obtained. At its inter- 
section with the indefinite perspective of the diagonal oa, locate 
A, the perspective of the apex. The horizontal projection of 
the point of sight is not shown as the location of v' from s' and v 
from s' also gives the distance of the point of sight from the pic- 
ture plane (1325). 



1329. Problem 5. To find the perspective of an arch. 

Construction. Fig. 248 shows plan and elevation of the 
arch. The corner nearest to the observer is in the picture plane, 
and therefore the lines in the plane are shown in their true length. 



PERSPECTIVE PROJECTION 



255 




00 

6 



256 CONVERGENT PROJECTING-LINE DRAWING 

This length need not be in actual size but may be dra^\Ti to any 
scale desired. From the extremities of these lines, others are 
dra\\Ti to the vanishing points Y' and V, as these are the van- 
ishing points of the perspectives of the parallels to the sides of 
the arch. 

It is to be remembered that through s, a line sm is drawn 
parallel to the reversed direction of op, because the plan would 
ordinarily be above the ground line. This convention may or 
may not be adopted. In Fig. 249 it is sho^^Ti.in another way 
and perhaps this may seem preferable. The matter is one of 
personal choice, however, and to the experienced, the hability to 
error is negligible. 

The most important feature of this problem is the location of 
points on the arch.* The advantage of the use of perpendiculars 
and diagonals, to locate the points A, B and C, may be shown 
here. Were these points located by finding the piercing points 
of the visual ray, the reader would soon find the number of lines 
most confusing, due to the lack of symmetry of direction; the 
use of the 45° and 60° triangles for the diagonals is much more 
convenient. 

In commercial perspective drawing, the draftsman estimates 
such small curves as are shown at the base. Only such points 
are located which are necessary for guides. In this case, 
the verticals and their limiting position are all that are 
required. 

1330. Problem 6. To find the perspective of a building. 

Construction. The plan and elevations of the building are 
shown on Fig. 249 draT\Ti to scale. In drawing the perspective, 
the plan only is necessary to locate the various lines. Continual 
reference must be made to the elevations for various points in 
the height of the building. Considerable work can be saved by 
having one corner of the building in the picture plane. Thus, 
that corner is shown in its true length even though it may 
not be the actual length. In the case shown, it is drawn to 
scale. 

The vanishing points of the horizontal lines on the building 
have been located from the horizontal projection of the point 

* This arch may also be drawn by craticulation. See Art. 1331. 



PERSPECTIVE PROJECTION 



257 




258 COXVERGEXT PROJECTING-LIXE DRAWIXG 

of sight, above the ground hne. The construction Hues are 
therefore drawTi parallel to the sides of the building. The dis- 
tances of the TVTQdow lines may be laid out to scale on the corner 
of the building which lies in the picture plane, as sho-^Ti. Strictly 
speaking, the term " scale " cannot be apphed to perspective 
drawing, as a line of a given length is projected as a shorter line 
as the distance from the observer increases. 

1331. Commercial application of perspective. The artistic 
requirement of perspective involves some choice in the selection 
of the point of sight. In general, the average observer's point 
of sight is about five feet above the ground, unless there is some 
reason to change it. The center of the picture should be chosen 
so that it is as nearh' as possible in the centre of gravity of area 
of the picture and this may modify the selection of the observer's 
point of sight. For instance, if a perspective of a sphere is 
made and the centre of the picture is chosen in the centre of 
graA^ty of the area, then the perspective is a circle. Other^-ise, 
if the centre of the picture is to one side, the perspective is an 
ellipse,* and, to properly view the picture, it should be held to 
one side. This is contrary to the usual custom; an observer 
holds the picture before him so that the average of the visual 
rays is normal to the plane of the paper. On building or similar 
work, the roof and base lines should not converge to an angle 
greater than about 50°. This can be overcome by increasing 
the distance between the vanishing points, or, what amounts to 
the same thing, increasing the distance between observer and 
object. 

It is also desirable to choose the position of the observer so as 
to show the most attractive view of the object most prominently. 
^Tiere a building has two equally prominent adjacent sides, 
it is a good plan to adjust the observer's position to present the 
sides approximately in proportion to their lengths. That is, 
when one side of a building is 200 feet long and 50 feet wide, 
the length of the building should appear on the drawing about 
four times as long as the width (these recommendations have 
been purposely ignored in Fig. 249). 

* The limiting rays from the sphere are elements of the sm-face of a 
cone of revolution; hence, its intersection with a plane inclined to its axis 
will be an ellipse. 



PERSPECTIVE PROJECTION 



259 



When impressions of magnitudes of objects are to be con- 
veyed, it can be done by placing men at various places on the 
drawing. A mental estimate of the height of a man will roughly 
give an estimate of the magnitude of the object. 

When curves are present in a drawing, the use of bounding 
figures of simple shape again finds application (210, 408). 
Architects know this under the name of craticulation. Fig. 250 




Fiu. 250. 



shows an example as applied to the drawing of a gas-ongine 
fly-wheel. A slightly better picture could have been obtained 
by giving the wheel a tilt so as to show the " section " more 
prominently. This would also bring the centre of the jiicture 
nearer to the centre of gravity of the area. If the reader will 
study a few photographs, kcn^ping these suggestions in mind he 
will note the conditions which make certain pictures better than 
others. 



260 CONYERGEXT PROJECTING-LINE DRAWING 

To make large perspectives is sometimes inconvenient due 
to the remoteness of the vanishing points. This requires a large 
drawing board and frequently the accuracy is impaired by the 
unwdeldiness of the necessary drawing instruments. To over- 
come these objections, it is possible to make a small drawing of 
the object and then to redraw to a larger one by m.eans of a pro- 
portional divider or by a pantograph. Only the more important 
lines need be located on the dra^^'ing. Such details as windows, 
doors, etc., can usually be estimated well enough so as to avoid 
suspicion as to their accuracy. The artist, with a little practice, 
soon accustoms himself to filling in detail. 

Freehand perspective sketches can be made with a little 
practice. . To make these, lay out the horizon and two vanish- 
ing points for building work and perhaps one or more for machine 
details. Accuracy is usually not a prerequisite, and, therefore, 
sketches of this kind can be made in almost the same time as 
oblique or axonometric sketches. The experience gained from 
this chapter should have furnished sufficient principles to be 
applied directly. 

The chief function of perspective is to present pictures to 
those unfamiliar wiih. dra^vdng. It is therefore desirable to use 
every effort possible to present the best possible picture from the 
artistic viewpoint. The increased tune required to make per- 
spectives is largely offset by the ease with which the uninitiated 
are able to read them. The largest application of perspective 
is found in architect's draT\dngs to chents, and in the making of 
artistic catalogue cuts. With perspective is usually associated 
the pictorial effect of illumination so as to call on the imagination 
to the minimum extent. 

1332. Classification of projections.* When drawings are 
made on a plane surface then there are two general sj^stems 
employed: those in which the projecting lines are parallel to 
each other and those in which the projecting lines converge to a 
point. 

* See Art. 413 in this connection. 



PERSPECTIVE PROJECTION 



261 











' Commonly 










called ortho- 








' Showing two 


graphic pro- 








dimensions 


jections or 
Mechanical 
. drawings. 






'Orthographic 


Showing three 
dimensions 


Isometric 




' Parallel 
projecting 




and known 
as axonome- 


Dimetric 




lines 




tric projec- 


Trimetric 


Projections 
on plane 




, Oblique 


tions. 




surfaces 


Convergent 
projecting 
lines 


\ OutKne alone is shown 
perspective I 

Aerial / Color and illumination added 






perspective 1 


. to Hnear perspective 



QUESTIONS ON CHAPTER XIII 

1. What is a scenographic projection? 

2. What is a linear perspective? 

3. Define visual rays. 

4. Define visual angle. 

5. What is the physiological effect of a variation in the visual angle? 

6. How does binocular vision afford a means of estimating distance? 

7. Why is the eye assumed as a single point in perspective? 

8. What is a vanishing point? Give example. 

9. Is the vanishing point the vanishing point of the line or of its per- 

spective? 

10. Show the theory of perspective by a window-pane illustration. 

11. What is an aerial perspective? 

12. What is the picture plane? 

13. How is the picture plane usually located with respect to the object 

and the observer? 

14. Show by an oblique projection how the perspective of a line is 

constructed. 

15. Show how the perspective of a line is constructed in orthograpliic 

projection. 

16. Prove that all perspectives of lines perpendicular to the horizontal 

plane have vertical perspectives. 

17. Prove that all perspectives of lines parallel to both ])riiicip;il pianos 

have perspectives j)arallel to the ground line. 

18. What is the centre of the picture? 

19. Prove that the perspectives of all lines perpendicular to the picture 

plane vanisli in the centre of the picture. 



262 CONVERGENT PROJECTING-LINE DRAWING 

20. Is the centre of the picture the vertical projection of the point of sight? 

21. What is a system of hnes? 

22. Prove that the perspectives of all systems of lines have a common 

vanishing point. 

23. How is the vanishing point of a system of hnes found? 

24. What is the horizon? 

25. Prove that the perspectives of all horizontal sj^stems of lines have 

a vanishing point on the horizon. 

26. Why is the centre of the picture on the horizon? 

27. Find the perspective of a point which is located in the second angle, 

by means of the piercing point of the visual ra}^ on the picture 
plane. 

28. Find the perspective of a point w^hich is located in the first angle, 

by means of the piercing point of the casual ray on the picture 
plane. 

29. Show that the vertical projection of a point which is situated in 

the picture plane is its owm perspective. 

30. Show that the vertical projection of a fine which is situated in the 

picture plane is its owm perspective. 

31. What is the indefinite perspective of a line? Give proof. 

32. Find the perspective of a cube hj means of the piercing pointsof 

the visual rays on the picture plane. 

33. Prove that if two lines in space intersect, their perspectives intersect 

in a point which is the perspective of the point of intersection 
on the lines. 

34. What is a perpendicular when apphed to perspective? 

35. Where does the perspective of a perpendicular vanish? 

36. What is a diagonal when applied to perspective? 

37. How many diagonals may be drawn through a given point? 

38. Where do the chagonals vanish? Why? 

39. What angle is generallj^ used for the diagonals? What other angles 

may be used? 

40. Find the perspective of a second angle point, b}^ the method of 

perpendiculars and diagonals. 

41. Find the perspective of a second angle point, by drawing two diagonals 

through it. Check by drawing the perpendicular. 

42. Find the perspective of a first angle point, by the method of per- 

pendiculars and diagonals. 

43. Find the perspective of a first angle point, by dramng two diagonals 

through it. Check by dramng the perpendicular. 

44. Find the perspective of a second angle fine, bj' the method of per- 

pendiculars and chagonals. 

45. Find the perspective of a second angle fine, by draT\dng the diagonals 

only. Check by drawing the perpendicular and visual rays. 

46. Find the perspective of a first angle fine, by the method of perpen- 

diculars and diagonals. 

47. Find the perspective of a first angle fine, by drawing the diagonals 

only. Check b}^ drawing the perpendiculars and \isual rays. 



PERSPECTIVE PROJECTION 263 

48. What is the object of revolving the horizontal plane in perspective? 

49. What precaution must be used in perspective with reference to the 

diagonals, when the horizontal plane is revolved? 

50. Find the perspective of a second angle point, by the method of 

perpendiculars and diagonals, when the horizontal plane is revolved. 

51. Find the perspective of a first angle point, by the method of per- 

pendiculars and diagonals, when the horizontal plane is revolved. 

52. Find the perspective of a second angle line, by the method of per- 

pendiculars and diagonals, when the horizontal plane is revolved. 

53. Find the perspective of a first angle line, by the method of perpen- 

, diculars and diagonals, when the horizontal plane is revolved. 

54. When the chagonals make an angle of 45° with the picture plane, 

how far are the vanishing points from the centre of the picture? 

55. Find the perspective of a cube by means of perpendiculars and 

diagonals. 

56. Why should the centre of the picture be chosen as near as possible 

to the centre of gravity of the perspective? 

57. What is the perspective of a sphere, when the centre of the picture 

is in the centre of gravity of the perspective? 

58. What is the perspective of a sphere, when the centre of the picture 

does not coincide with the centre of gravity of the perspective? 

59. What should be the approximate angle of convergence of the roof 

and base Unes of a building on a perspective? 

60. If the roof and base lines of a building converge to an angle con- 

sidered too large, what remedy is there for this condition? 

61. When adjacent sides of a building are equally attractive, what should 

be their general relation on the perspective? 

62. How may impressions of magnitude be conveyed on a perspective? 

63. What is craticulation? 

64. How may large perspectives be made when the drawing board is 

small? 

65. When perspective is commercially applied, is it necessary to locate 

every point on the details or can it be drawn with sufficient 
accurac}^ by estimation? 

66. Make a complete classification of projections having parallel and 

convergent projecting lines. 

Note. In the following drawings, keep the centre of the picture as 
near as possible to the centre of gravity of the drawing. 

67. Make a perspective of Fig. 1 in text. 

68. Make a perspective of Fig. 10 in text. 

69. Make a perspective of Fig. 17 in text. 

70. Make a perspective of Fig, 2A. (Question in Chap. II.) 

71. Make a perspective of Fig, 2B. (Question in Chap. II.) 

72. Make a perspective of Fig, 21 in text. 

73. Make a perspective of Fig, 3.1. (Question in Chap. III.) 

74. Make a perspective of Fig, SB. " " " 

75. Make a perspective of Fig. 3C. " " " 



264 



CONVERGENT PROJECTI>Xt-LINE DRAWING 



(Question in Chap. III.) 



76. Make a perspective of Fig. SD. 

77. Make a perspective of Fig. SE. 

78. Make a perspective of Fig. 3F. 

79. !Make a perspective of Fig. 3G. 
SO. Make a perspective of Fig. 3^. 

81. Make a perspective of Fig. 3/. 

82. Make a perspective of Fig. 3-/. 

83. Make a perspective of Fig. SK. 

84. Make a perspective of Fig. 3L. 

So. Make a perspective of Fig. 40 in text. 

86. Make a perspective of Fig. 41 in text. 

87. Make a perspective of Fig. 42 in text. 

88. Make a perspective of a pyramid on a square base. 

89. Make a perspective of an arch. 

90. Make a perspective of a building. 



PART IV 
PICTORIAL EFFECTS OF ILLUMINATION 



CHAPTER XIV 

PICTORIAL EFFECTS OF ILLUMINATION IN ORTHOGRAPHIC 

PROJECTION 



1401. Introductory. The phenomena of illumination on 
objects in space is a branch of the science of engineering drawing 
which aims to give the observer a correct imitation of the effect 
of light on the appearance of an object. In the main, it is desired 
to picture reality. The underlying principles of illumination 
are taken from that branch of physics known as Optics or Light. 
The application of these principles to graphical presentation 
properly forms a part of drawing. 

It is needless to say that the subject borders on the artistic; 
yet it is sometimes desirable to present pictures to those who are 
unfamiliar mth the reading of commercial orthographic pro- 
jections. The architect takes advantage of perspective in con- 
nection with the effects of illumination, and in this way brings 
out striking contrasts and forcibly attracts attention to the idea 
expressed in the drawing. 

It is not always essential, however, to bring out striking 
effects, as occasion often arises to picture the surfaces of an 
object. This is done by pic- 
torially representing the effects 
of illumination. 



1402. Line shading applied 
to straight lines. The simplest 
application of line shading is 
shown in Fig. 251; the object 
is a rectangular cover for a box. In the illumination. 



mm 



mii 



I-IG. 251. 



it is assumed 
265 



266 



PICTORIAL EFFECTS OF ILLUMINATION 



mm 



that the Hght comes in parallel lines, from the upper left-hand 
corner of the drawing. The direction is downward, and to the 

right, at an angle of 45°. The 

illuminated surfaces are drawn 

^vith any thickness (or, as is 

commercially termed, '^ weight ") 

of line; the surfaces not in the 

light are made heavier as though 

Fig. 251. the line or surface actually cast 

a shadow. Irrespective of the 

location on the sheet, the drawing would always be represented 

in the same way since the light is assumed to come in parallel 

lines. 

Evidently there is (strictly speaking) no underlying theory 
in this mode of shading, the process being merely a convention 
adopted by draftsmen ; thus its extended use merely is the neces- 
sary recommendation for its value. 

1403. Line shading applied to curved lines. In Fig. 252 




Fig. 252. 



is shown one part of a flange coupling, the illustration being 
chosen to show the application of line shading to the drawing of 
curves. In this latter application all is given that is necessary 
to make drawings using this mode of representat'on. 

As before, the light comes from the upper left-hand comer 



IN ORTHOGRAPHIC PROJECTION 267 

of the drawing. By the aid of the two views (one-half of one 
being shown in section) such surfaces as will cast a shadow are 
easily distinguished. The shade lines of the circles are shown 
to taper gradually off to a diagonal line ab. In drawing the shade 
line of a circle, for instance, draw the circle with the weight of 
line adopted in making the drawing; with the same radius and 
a centre located shghtly eccentric (as shown much exaggerated 
at c in Fig. 252), draw another semicircle, adding thickness to 
one side of the circle or the other, depending upon whether it is 
illuminated or not. 

It will be seen that this second circle \vill intersect the first 
somewhere near the diagonal ab. For instance, the extreme 
outside circle casts a shadow on the lower right-hand side, whereas 
the next circle is shaded on the upper left-hand side. These 
two shaded circles indicate that there is a projection on the 
surface. In the same manner the drawing is completed. It 
makes no difference whether the surface projects much or little, 
the lines are all shaded in the same way and with the same weight 
of shade line. 

Notice should be taken that for all concentric circles, the 
eccentric centre is always the same. The six bolt holes are 
also shown shaded, and each of which has its own eccentric 
ce:tre. 

Many cases arise in practice where there are projec- 
tions or depressions in the surface. This convention helps to 
interpret, rapidly and correctly, such drawings. The time 
taken to use this method is more than offset by the advantages 
thereby derived ; only in extremely simple drawings does it become 
unnecessary. 

1404. Line shading applied to sections. It makes no dif- 
ference whether the outside view is shown, or whether the 
object is shown in section (Fig. 252) the shade lines are 
drawn as though the supplementary planes actually out the 
object so as to expose that portion. This is done for uniformity 
only. 

1405. Line shading appHed to convex surfaces. Occa- 
sionally, curvcMl surfa('(\s must be contrasted with flat surfaces, or, 
perhaps, the effect of curvature l)n)ught(>ut without an atti^mpt to 



268 



PICTORIAL EFFECTS OF ILLUMINATION 



contrast it with flat surfaces. Fig. 253 shows a cyUnder shaded. 
The heavier shade is shown to the right as the Ught is supposed 
to come from the left; the reason for this will be shown later. 
The effect of shading and its graduation is produced by gradually 
altering the space between the lines, however, keeping the weight 
of hne the same. A somewhat better effect is produced by also 
increasing the weight of the line in connection with the decrease 
in the spacing, but the custom is not general. 

1406. Line shading applied to concave surfaces. A hollow 
cyhnder is shown shaded in Fig. 254. Again, the light comes 
from the left and the heavier shade falls on the left, as shown. 



11 




Fig. 253. 



Fig. 254. 



A comparison between Figs. 253 and 254 shows how concave and 
convex surfaces can be indicated. 

1407. Line shading applied to plane surfaces. Flat 
surfaces are sometimes shaded by spacing lines an equal distance 
apart. The effect produced is that of a flat shade. The method 
is used when several surfaces are shown in one view whose planes 
make different angles with the planes of projection, like an 
octagonal prism, for instance. 

PHYSICAL PRINCIPLES OF LIGHT . 

1408. Physiological effect of light. Objects are made 
evident to us by the reflected light they send to our eyes. The 
brain becomes conscious of the form of the object due to the 
physiological effect of hght on the retina of the eye. The locality 
from which the light emanates is called the source. Light travels 
in straight lines, called rays, unless obstructed by an opaque 
body. If the source of light is at a considerable distance from 
the object, the light can be assumed to travel in parallel lines. 



H 



IN ORTHOGRAPHIC PROJECTION 269 

The chief source of light is the sun and its distance is so great 
(about 93,000,000 miles) that all rays are practically parallel. 

1409. Conventional direction of light rays. The source 
of light may be located anywhere, but it is usual to assume that 
it comes from over the left shoulder of the observer who is viewing 
an object before him. The projections of the rays on the hor- 
izontal and vertical plane make an angle of 45° with the ground 
line. The problems to follow will be worked out on this assumed 
basis. 

1410. Shade and shadow. The part of the object that is 
not illuminated by direct rays is termed the shade. The area 
from which light is excluded, whether on the object itself or on 
any other surface, is called the shadow. As an example, take a 
building with the sun on one side; the opposite side is evidently 
in the shade. A cornice on this building may cast a shadow 
on the walls of the building, while the entire structure casts a 
shadow on the ground, and sometimes on neighboring buildings. 

1411. Umbra and penumbra. When the source of light 
is chosen near the object, two distinct shadows are observable, 
one within the other. Fig. 255 shows a plan view of a flat gas 
flame ab and a card cd. Rays emanate from every point of the 
flame in all directions. Consider the point a, in the flame. 
The two rays ac and ad will determine a 

shadow cast by the card the area of which 
is located away from the flame and limited 
to that behind fcdh. Similarly, from the 
point b, the two rays be and bd cast a 
shadow, limited by the area behind ecdg. 
These two areas overlap. From the posi- 
tion of fcdg, no part of the flame can ])e 
seen by an observer standing there; but 
from any one of the areas behind ecf or gdh, 
a portion of the flame can be seen. The 

effect of this is that the shadow in fcdg is much more i)rc)nouni'ed 
and therefore is the darker. 

That portion of the shadow from which the ligiit is totally 
excluded (fcdg) is callcMl tlu^ umbra; that portion from wliicli 
the light is partly excluded (ecf or gdh) is called the penumbra. 




270 



PICTORIAL EFFECTS OF ILLUMINATION 



This effect is easily shown by experiment, and is sometimes 
seen in photographs taken by artificial illumination when the 
source of light is quite near the object. In passing, it may also 
be mentioned that if there are two sources of light, there may be 
two distinct shadows, each having its own umbra and penumbra, 
but such cases are not usually considered in drawing. 

When applying these principles to drawing, the shadow 
is limited to the umbra, and the penumbra is entirely neglected. 
The light is supposed to come from an infinite distance, and, 
therefore, in parallel rays. So little in the total effect of a draw- 
ing is lost by making these assumptions, that the extra labor 
involved in making theoretically correct shadows is uncalled for. 

GRAPHICAL REPRESENTATION 

1412. Application of the physical principles of light to 
drawing. In the application of the physical principles enumerated 
above, it is merely necessary to draw the rays of light to the 
limiting lines of the object, and find their piercing points with 
the surface on which the shadow is cast. This is the entire 
theory. 

1413. Shadows of lines.* Fig. 256 exhibits the shadow cast 
by an arrow AB shown by its horizontal projection ab and its 






Fig. 256. 



Fig. 257. 



vertical projection a'b'. The rays of light come, as assumed, 

* Lines are mathematical concepts and have no width or thickness. 
They can, therefore, strictly speaking, cast no shadow. 



IN ORTHOGRAPHIC PROJECTION 271 

in lines whose horizontal and vertical projections make an angle 
of 45° with the ground line. A ray of light to A is shown as ca 
in the horizontal projection and c'a' in the vertical projection. 
The ray of light to B is shown by the similar projection db 
and d'b'. These two rays pierce the horizontal plane at a" 
and b'' and a''b'' is therefore the shadow of the arrow AB in 
space. 

If the arrow is as shown in Fig. 257, then the shadow is on the 
vertical plane. A similar method is used for drawing the pro- 
jections of the rays on the horizontal and vertical planes. Instead, 
however, the piercing points on the vertical plane are found, 
and these are shown as a'^b'', which again is the shadow of AB 
in space. 

In Fig. 256 the rays pierce the horizontal plane before they 
pierce the vertical plane and the shadow is therefore on the 
horizontal plane. Since the plane is assumed opaque there is no 
vertical shadow. Fig. 257 is the reverse of this. A vertical 
shadow and no horizontal shadow is 
obtained. Cases may occur where 
the shadow is partly on both planes, 
similar to a shadow cast partly on the 
floor and partly on the wall of a room. 
Fig. 258 shows this in construction. 
It differs from the preceding in so far 
as the two horizontal and the two ver- 
tical piercing points of the rays are Fig. 258. 
obtained, and thus the direction of the 

shadow is determined on both planes. If the construction is 
carried out accurately, the projections will intersect at the ground 
line, and the shadow will appear shown as a"b''. 

1414. Problem 1. To find the shadow cast by a cube which 
rests on a plane. 

Let, in Fig. 259, abed be the horizontal projection of the top 
of the cube and a'b'c'd' be the vertical projection of the top of 
the cube. The three points in space A, B and C will bo sullicitMit 
to determine the shadow and, as a consequence, the projections 
of these rays are drawn at an angle of 45°, as shown. The lior- 
izontal projections of the rays are accordingly ae, bf and eg; 
the vertical projections are likewise a'e', h't' and c'g'. The 




272 



PICTORIAL EFFECTS OF ILLUMINATION 



piercing points (506, 805) of these rays are e, f and g. The 
shadow i^ therefore determined because the cube rests on the 
horizontal plane, and the base must necessarily be its own 
shadow. 

1415. Problem 2. To find tiie shadow cast, by a pyramid, 
on the principal planes. 

In order to show a case where the shadow is cast on both 
planes, the pyramid must be located close to the vertical plane. 
The pyramid, ABCDE, Fig. 260, is shown in the customary way 




Fig. 259 



as abcde or the horizontal projection, and a'b'c'd'e' for the 
vertical projection. The horizontal piercing point of what would 
be the shadow of the apex must be located in order to determine 
the shadow of the side BA, even though it forms no part of the 
actual shadow. Locate also the vertical piercing point g' which 
is the actual shadow of the apex. It will be noticed that the 
shadow of BA only continues until it meets the vertical plane and 
is therefore limited by the projection bh. Similarly, the shadow 
is limited by di, a portion of the line df. Where these two lines 
meet the ground line, join the points with the actual shadow 
of the apex an-d the shadow is then completed. It may be 



IN ORTHOGRAPHIC PROJECTION 



273 



observed that the Umiting Hnes of an object determine the 
shadow in all cases, and that the interior lines do not influence 
the figure unless they project above the rest of the object in any 
way. 

1416. Problem 3. To find the shade and shadow cast by 
an octagonal prism having a superimposed octagonal cap. 

Let Fig. 261 represent the object in question. Confine atten- 
tion at present to the shadow on the horizontal plane. Con- 



c'fe' d'a' 




do y ^^ 
Fig. 261. 



sider, first, the shadow cast by the lino GH in space. It will 
be seen that the rays of light from G and H pierce the horizontal 
plane at points g" and h" and that these points are found by 
drawing the projections of the rays from the points G and H in 
space. If the construction is correct so far, then g"h" should 
be parallel to gh, as GH is parallel to the horizontal i)lane and as 
its shadow must be parallel both to the line itself and also to 
its horizontal projection. Similarly, locate a" and f". Thus, 
three linos of the shadow are dotorminod. 

It is now nocossarv to observe oarofuUv that (ho sliadow of 



274 



PICTORIAL EFFECTS OF ILLUMINATION 



the superimposed cap is not only due to the top face, as the points 
jj^//j^//^//p// ^j.g ^YiQ piercing points of rays that come from the 
points MNOP, in space, and that they are on the lower face of 
the cap. The shadow of the points BCDE fall within the area 
8L"h"g"f'p''o"m"n" and therefore need not be located. The 
shadow is completed by drawing r"s'' and t''u'' which are a part 
of the shadow cast by the octagonal prism below. It is unneces- 
sary to determine the complete outline of the shadow of the 




do y ^^ 



Fig. 261. 



prism as it merges into that cast by the cap. The shadow on 
the horizontal plane is thus completely determined. 

Consider, in addition, the shadow cast on the prism due to 
the superimposed cap. The shadow limited by the line v'V 
is due to the limited portion of the lower face of the cap shown 
horizontally projected at vw and vertically as v'w^ The shadow 
of the point O in space is shown as x'^ and that of Y in space as 
y". All these latter points are joined by straight lines as they 
are shadows of straight lines cast upon a plane surface and may 
be considered as the intersection of a plane of rays through the 
line forming the outline of the object. 



IN ORTHOGRAPHIC PROJECTION 



275 



The right of the face of the prism and of the cap is shaded 
entirely, because it is in the shade, and since it receives no direct 
illumination, it is represented as shown. 



1417. Problem 4. To find the shade and shadow cast 
by a superimposed circular cap on a cylinder. 

Fig. 262 shows the cylinder with its superimposed cap. The 
shadow cast by the cap will be some form of curve, and attention 
will be directed at present, to the location points on this curve. 
From a', in the vertical projection, draw a ray whose projection 
makes an angle of 45° with the ground 
line, and locate it so that it impinges on 
the surface of the cylinder at the extreme 
visible element to the left. The corre- 
sponding horizontal projection of a' is a, 
and the point where this ray impinges is 
a/' one point of the required shadow. 
Consider point b, vertically projected 
at b'. The projections of its ray will in- 
tersect the surface of the cylinder at the 
point b'' along an element shown ver- 
tically projected at h'". Select another 
point c' in the vertical projection, so that 
the ray c'd' will be tangent to the cyl- 
inder. The element is horizontally pro- 
jected at c"d and the ray impinges at c" 
which is still another point of the shadow. 

In practice, several more points are determined, but they are 
omitted here for the sake of clearness. The portion of the 
cylinder away from the source of light must be in the shade 
and d' shows the vertical projection of one element from which 
the illuminated portion is separated from the shade. Accord- 
ingly, the portion of the cylinder to the right of c"d is shaded 
entirely. The same effect occurs on the superimposed cap. and, 
hence, from the element e, to the right, the (mtire remaining art^a 
is shaded, being unilluminated from rays having projections which 
make an angle 45° with the ground line. 




Fig. 202 



1418. Iiigh=light. When a body, such as a polisluul sphere, 
is subjected to a source of light, one spot on the sphere will appear 



276 PICTORIAL EFFECTS OF ILLUMINATION 

much brighter than the rest of the sphere. This spot is called 
the brilliant point, or, more commonly, the high-light. The 
eJBFect is that due to the light being immediately reflected to the 
eye with practically undiminished intensitj'. 

1419. Incident and reflected rays. It is a principle in 
optics * that the incident ray and the reflected ray make equal 
angles with the normal to the surface on which the light impinges. 
The tangent plane at the point where the light impinges must be 
perpendicular to the normal; the incident ra^^ and the reflected 
ray also lie in the same plane with the normal and therefore 
this plane is perpendicular to the tangent plane. It may be 
observed that if a line be draT\Ti through am^ point in space 
parallel to the ray of Hght or incident ray, and through the 
same point, another line be dra\^TL parallel to the reflected ray, 
then the included angle will be the same irrespective of the loca- 
tion of the arbitrar}^ point chosen. The normal through this 
point would therefore be parallel to a normal dra^Ti through any 
other point in space, and a plane perpendicular to one normal 
would be perpendicular to all. By choosing the perpendicular 
plane so that it touches the surface a tangent plane to the sur- 
face is obtained; the point of contact is then the high-light. 
The foregoing can best be illustrated by an example. 

1420. Problem 5. To find the high-light on a sphere. 

To use a simple illustration, the high-light on the vertical 

projection only will be deter- 
"^ V mined. Fig. 263 shows this con- 

^^^<\ ^yf^ struction. Assume that the eye 

''yf^^^A / ^^'^ ^^ directed perpendicularly to the 

/ \\ X / y/\ vertical plane. The light comes 

/ ! ^\ \ / \ ^^ ^ direction whose projections 

I 1^ ^^ L niake an angle of 45° \^ith the 

I / ground line. For convenience, 

\ / take a ray through the centre 

\ / of the sphere o'; the incident 

^ ^y^ ray is therefore m'o'. The Une 

Pig. 263. through the point of sight is per- 

pendicular to the plane of the 
paper, and is therefore projected at o'. Revolve the plane of 

* See a text-book on Phvsics, 



IN ORTHOGRAPHIC PROJECTION 277 

these incident and reflected rays about m'o' until it coin- 
cides or is parallel with the vertical plane. The perpendic- 
ular to the plane of the paper through o' will fall to c" 
and the revolved angle will be c''o'm''; the angle m'o'm'' 
being numerically * equal to 35° 16' for rays whose projections 
make an angle of 45° with the ground Une. The point m'' is 
graphically located as follows: Assume that a horizontal plane 
is passed through o', the centre of the sphere. The distance 
of the point va! from this new horizontal plane is m'q' and this 
distance does not change in the revolution. Hence, the point M 
in space falls at m'', cm'' being the revolved position of the actual 
ray of light. 

The bisector of the angle m^'o'c'' is the normal to the surface 
and a plane tangent to the revolved position of the sphere will 
be tangent at the point p". It is unnecessary to draw the plane 
in this case as the normal readily determines p''. The counter 
revolved position of p" is p' ; therefore, p' is the required high- 
hght. 

1421. Multiple high=lights. Before leaving this subject it 
must be noted that with several sources of light on an object, 
there may be several high-lights, although it is usual to assume 
only one on a surface like the sphere. A corrugated surface will 
have several high-lights for any one position of the eye and a 
single source of light. 

1422. High=lights of cylindrical or conical surfaces. A 

cylinder or a cone can have only one high-light for an}' one posi- 
tion and direction of vision of the eye. If the surface is highly 
polished, it will be the only point visible; the object itself cannot 
be distinguished as to the details of its form. As the eye is 
directed anywhere along the surface, a new high-light is observed 
for each direction of vision and the locus of these points forms 
approximately a straight line. 

1423. Aerial effect of illumination. The foregoing j)riiu'i- 
plcs are true for the conditions assumed, lu nature, however, it is 
impossible to receive light from only one source in directly parallel 
rays, to the total exclusion of any other light. Modifications 
must be introduced because the surfaces are not optically true 

* Coinputcd by trigonomiMrv. 



278 PICTORIAL EFFECTS OF ILLUMINATION 

in the first instance. This means that a cyUnder, although made 
round as carefully as possible, very slight deviations hardly 
discernible by measuring instruments are readily detected by 
their appearance in the light. The reflected light augments 
the imperfections, so that to the experienced eye the effect is 
noticeable. Other equally important facts are those due to the 
reflection from the walls of the room, and from other objects; 
the diffusion caused by the light being transmitted through the 
window-pane, and that reflected from dust particles suspended 
in the air. All these disturbing influences tend to illuminate 
certain other portions of the object and actually do so to such 
an extent that cognizance of this aerial effect of illumination 
must be taken. 

1424. Graduation of shade. The effect on a highly pohshed 
sphere can be brought out b}^ holding one in a room whose walls 
are a dull black, similar to a room used for photometric testing. 
Even this photometer room does not fulfil the requirements to 
the last degree, as there is no siu-face which does not reflect at 
least some light. Observing this sphere by aid of a ray of light 
coming through any opening in the room, the high-light will 
alone be visible. If there were no variation in the amount of 
light sent to the eye from the various points on the object, due 
to the aerial effect of illumination, it would be impossible to 
recognize its form. Hence no light can come to the eye except 
from the high-light unless the aerial effect is present. 

With these difficulties to contend T\dth, it may seem that the 
foregoing principles become invalid. Such is not the case, 
however, as the principles are true, in the main, but the proper 
correction is made for diffused light and a graduation in shade 
is introduced so as make the observer conscious of the desired 
idea to be conveyed. 

1425. Shading rules. At times it may even be necessary to 
strain a point in order to bring out some detail of the object 
drawn. This latter is a characteristic of the skill of the artist. 
The rules for the representation of shades are here inserted, being 
taken from Mahan's Industrial Drawing: 

''1. Flat tints should be given to plane surfaces, when in the 
light, and parallel to the vertical plane; those nearest the eye 
being lightest. 





Fig. 264. 



Fig. 265. 





Fig. 266. Fig. 267. 

Examples of Graduated Sliades in Pictorial I'^UVcl.s of lUuuHnation. 

{To face paur ^7S] 



IN ORTHOGRAPHIC PROJECTION 279 

" 2. Flat tints should be given plane surfaces, when in the 
shade, and parallel to the vertical plane; those nearest the eye 
being darkest. 

''3. Graduated tints should be given to plane surfaces, when 
in the light and inclined to the vertical plane ; increasing the shade 
as the surfaces recede from the eye; when two such surfaces 
incline unequally the one on which the light falls most directly 
should be lightest. 

" 4. Graduated tints should be given to plane surfaces, when 
in the shade, and inclined to the vertical plane; decreasing the 
shade as the surfaces recede from the eye." 

1426. Examples of graduated shades. In Figs. 264, 265, 266 
and 267 are shown a prism, cylinder, cone and sphere shaded in 
accordance with the above rules. It will be seen that no mistake 
can occur as to the nature of the objects, thus certifying to the 
advisability of their adoption. 

QUESTIONS ON CHAPTER XIV 

1. What is the purpose of introducing the pictorial effects of illumina- 

tion? 

2. What is meant by line shading? 

3. How are the straight lines of an object line shaded? Give example. 

4. How are the curved Unes of an object Hne shaded? Give example. 

5. How arc the "sections" hne shaded? Give example. 

6. How arc convex surfaces line shaded? Give example. 

7. How arc concave surfaces line shaded? Give example. 

8. How are plane surfaces line shaded? Give example. 

9. How are objects made evident to us? 

10. What is the source of light? 

11. What are light rays? 

12. What conventional direction of ray is adopted in shading drawings? 

13. What is the shade on an object? 

14. What is the shadow of an object? 

15. What is the umbra? Give example. 

16. What is the penum])ra? Give example. 

17. Why is only the umbra used on drawings? 

18. What is the fundamental operation of finding the shadow of an 

object? 

19. Construct the shadow of a line (hat is situaUnl so as to hav(> a 

shadow on the horizontal ])la!ie. 

20. Construct the .shadow of a line that is situated so as to iiave a 

shadow on the vertical plane*. 



280 PICTORIAL EFFECTS OF ILLUMINATION 

21. Construct the shadow of a line that is situated so as to have a 

shadow on both principal planes. 

22. What is a high-light? Explain fully. 

23. What is the incident ray? 

24. What is the reflected ray? 

25. What angular relation is there between the normal and the incident 

and reflected rays? 

26. Do the normal, the incident ray and the reflected ray lie in one 

plane? 

27. Find the high-light on a sphere. 

28. How are multiple high-hghts produced? 

29. What is the aerial effect of illumination? 

30. What is meant by graduation of shade? 

31. To what is the graduation of shade due? 

32. Shade a sphere T\dth lead pencil so as to show the high-light and 

the graduation of the shade. 

33. Shade a cylinder with lead pencil so as to show the high-light and 

the graduation of the shade. 

34. Shade a cone with lead pencil so as to show the high-hght and the 

graduation of the shade. 

35. Shade an octagonal prism with lead pencil so as to show the high- 

light and the graduation of the shade. 

36. Construct the horizontal shadow of a cube resting on the horizontal 

plane. 

37. Construct the horizontal shadow of a cube which is some distance 

above the horizontal plane. 

38. Construct the horizontal shadow of a triangular prism which rests 

on the horizontal plane. 

39. Construct the horizontal shadow of a hexagonal prism which rests 

on the horizontal plane. 

40. Construct the horizontal shadow of a pyramid which rests on the 

horizontal plane. 

41. Construct the shade and horizontal shadow cast by a cylinder with 

a superimposed circular cap. 

42. Construct the shade and horizontal shadow cast by a cjiinder with 

a superimposed square cap. 

43. Construct the shade and horizontal shadow cast by a'cylinder with 

a superimposed octagonal cap. 

44. Construct the shade and horizontal shadow cast bj^ an octagonal 

prism with a superimposed circular cap. 

45. Construct the shade and horizontal shadow cast by an octagonal 

prism with a superimposed square cap. 

46. Construct the shade and horizontal shadow cast bj^ an octagonal 

prism with a superimposed octagonal cap. 

47. Construct the shade and horizontal shadow cast by a cone which 

rests on a square base. 

48. Construct the shade and horizontal shadow cast by an octagonal 

pyramid which rests on a square base. 



IN ORTHOGRAPHIC PROJECTION 



281 



49. Construct the shadow cast by a cone which is situated so as to 

cast a shadow on both principal planes. 

50. Construct the shadow cast by an octagonal pyramid which is situated 

so as to cast a shadow on both principal planes. 

51. Construct the shadow cast by an octagonal prism which is situated 

so as to cast a shadow on both principal planes. 

52. Construct the shadow cast by a superimposed cap on the inside 

of a hollow semi-cyhnder. 



CHAPTER XV 

PICTORIAL EFFECTS OF ILLUMINATION IN PERSPECTIVE 
PROJECTION 



1501. Introductory. The fundamental principles of the pic- 
torial effects of illumination are best studied in orthographic 
projection. Their ultimate use, however, is us rally associated 
with perspective. Thus, in one color, an attempt is made to 
picture reality, whether it be used for engineering purposes, as 
catalogue illustrations, or whether it be used for general illustrating 
purposes. The principles, established here, hold equally well 
when color is added to the perspective (making it an aerial per- 
spective) and its illumination; but, as this is recognized as a 
distinct field, it will not be considered in this book. 

1502. Problem 1. To draw the perspective of a rectangular 
prism and its shadow on the horizontal plane. 

The first step in this case is to draw the object and its shadow 
orthographically. The object is shown in the second angle as 

has been the custom in perspec- 
tive; the shadow is on the hori- 
zontal plane and its construc- 
tion is carried out in accord- 
ance with principles previously 
discussed (Chap. XIV). The 
shaded area in Fig. 268 shows 
the shadow so constructed. The 
advisability of the 180° revolu- 
tion of the horizontal plane has 
been given (1322), and carrying 
this into effect in the present 
instance, the effect shown in 
Fig. 269 is obtained. The hori- 
zontal projection of the prism and the shadow of the prism are 
then shown below the ground line. The perspective of the prism 

282 




Fig. 268. 



IN PERSPECTIVE PROJECTION 



283 



will perhaps be clear from the illustration in Fig. 269 as all the 
necessary construction hnes have been included. The steps 
necessary for the construction of the perspective of the shadow, 
however, will be considered. 

The point of sight is at S, shown horizontally projected at 
s and vertically projected at s\ The vanishing points of the 
diagonals are shown as v and v' ; the distance of v and v' from 
s' are equal to the distance s above the ground line (1325). A 




Fig. 269. 



perpendicular and a diagonal through the point d'' (the shadow 
of D in space) will intersect at d'" which is the required perspootivo 
of the shadow of this point. The same procedure will locate 
c'" and W. The corners F and G are the pers]-)(H'tives of those 
points in space, and as they rest on the horizontal piano, they 
are also the perspectives of their shadows. By joining Fb'"c'"d"' 
and H with lines the complete outline of tlu^ shadow is obtniiUHl, 
except in so far as the limited portion Ix^hind tlu^ prism froiu 
H which is hidden from the observer is concerned. 



284 



PICTORIAL EFFECTS OF ILLUMINATION 



1503. General method of finding the perspective of a 
shadow.* The above method of constructing the perspectives 
of shadows is perfectly general, although lengthy. It is possible 

to economize time, however, by taking advantage of the method 
of locating the perspective of the shadow directly. The shadow 
of a point in space on any surface is the piercing point of a ray 
of light through the point on that surface; the perspective of that 
shadow must therefore lie somewhere on the perspective of the 
ray of Hght. It will also lie on the line of intersection of the 
plane receiving the shadow with a plane containing the ray 
of light. The perspective of this line of intersection will also 




Fig. 270. 

contain the perspective of the shadow. Hence, the perspective 
of the shadow of the point will lie on the intersection of these 
two perspectives (1318). 

1504. Perspectives of parallel rays of light. The rajs 
of light are assumed as coming in parallel lines; being parallels, 
they therefore have a common vanishing point. To find this 
vanishing point (1313) draw through the point of sight, a line 
parallel to these rays; and the piercing point of this line on the 
picture plane will be the required vanishing point. In Fig. 
270, if a ray be drawn through the point of sight, then the vertical 

* This method is similar, in general, to the finding of the piercing point 
of a given Hne on a given plane. See Art. 823. 



n 



a 



IN PERSPECTIVE PROJECTION 285 

projection of the ray will be s'r'; the horizontal projection of 
the ray wiU be sr (a careful note being made of the 180° revo- 
lution of this plane and hence the revolved direction of the ray) 
and the piercing point on the picture plane will therefore be 
atr'. 

1505. Perspective of the intersection of the visual plane 
on the plane receiving the shadow. The line of intersection 
of the plane receiving the shadow and the plane containing the 
ray of light (or visual plane as this latter plane is called) is in 
our case a horizontal line, as the plane receiving the shadow is 
the horizontal plane. If the horizontal projecting plane of the 
ray be taken, it is known that the trace makes an angle of 45° 
with the ground line, and that this horizontal line miist vanish 
in the horizon at v. It will be observed that this same point 
(v) is also the vanishing point of all diagonals drawn to the 
right of the point of sight. A further note may be taken of the 
fact that if the perspective of the horizontal projection of the 
point be joined with the right vanishing point of the diagonal, 
the perspective is identical with the perspective of the intersection 
of the horizontal plane and the visual plane, because the per- 
spectives of the horizontal projection of the point and the van- 
ishing point are common to the two. 

1506. Application of the general method of finding the 
perspective of a shadow. The foregoing can be applied to the 
finding of the shadow of Problem 1. A reference to Fig. 270 
in addition to what follows, will indicate the application. Sui")]K)so 
the perspective of the shadow of D is under consideration. The 
perspective of the visual ray is Dr'; the perspective of the hori- 
zontal projection of the ray is Hv; their intersection is d'", 
the perspective sought. Likewise, c'^' is similarly located, and 
it is the perspective of the shadow of C, found by the inter- 
section of the Cr' (persi)ective of the ray) and Gv (p(m-sj)(H'- 
tive of the horizontal projection of the ray). The point B 
has its shadow at V, and its location is clearly shown in the 
figure. 

The shadow is completed by joining tiie proper points witli 
lines. In every way it is identical with the shadow determined 
in Problem 1. 



286 



PICTORIAL EFFECTS OF ILLUMINATION 



1507. Problem 2. To draw the perspective of an obelisk 
with its shade and shadow. 

Let AFGHK be the obelisk (Fig. 271) and S the point of 
sight. The perspective is drawn in the usual way. The problem 
is of interest in so far as the rays of light make an angle of 30° 
with the ground hne, thus causing a longer shadow than when 
the 45° ray is used. 

Through the horizontal and vertical projections of the point 
of sight, draw a ray parallel to the conventional ray adopted. 
This ray pierces the picture plane at r' the vanishing point of all 
the rays in space. The horizontal projections of all rays vanish 




Fig. 271. 



at r'' on the horizon, as all lines should that are parallel to the 
horizontal plane and at the same time belong to the system 
of lines parallel to the rays of light. 

Inspection of Fig. 271 will show that the lines GC, CA, AE, 
and EK affect the. shadow, and that the only points to be located 
for the shadow are C, A, and E. To locate c", the shadow of 
point C, draw the perspective of the ray through C; Cr' is this 
perspective. The perspective of the horizontal projection is 
d"i" \ d" is the perspective of the horizontal projection of C, 
the necessary construction lines being shown in the figure. The 
intersection of these two perspectives is c'', the required per- 
spective of the shadow of C in space. 

The point A has the perspective of its shadow at a" which. 




Fig. 273. — Commercial Application of the Pictorial Effects of Illumination in Perspective. 

[To face page £87] 



IN PERSPECTIVE PROJECTION 287 

as before, is the intersection of the perspective Ar' (of the ray) 
and 2i"x" (of the horizontal projection of the ray). The point 
E has its shadow q" located in an identical manner as the 
preceding points c" and a". 

As the obelisk is resting on a plane (the horizontal in this 
case) the base is its own shadow, and it is only necessary to join 
the shadow of C with G and the line of the shadow d'Qr is deter- 
mined. Likewise, join z" with a'', a'' with e'', and, finally, 
e'' with K. 

1508. Commercial application of the pictorial effects of 
illumination in perspective. A few, general remarks, in cases 
where the shadow falls on itself or nearby objects, may not be 
amiss. The draftsman usually has some choice in the selection 
of the direction of the rays, and, sometimes, in the location of 
nearby objects. 

Where the shadow is cast on the object itself or on neighboring 
objects, it wdll, in general, be found much easier to find the shadow, 
orthographically, and then to proceed with the making of a 
perspective from it. When the shadow is cast on a horizontal 
surface only, the general method outlined in Arts. 1503, 1504, 
and 1505 will find ready application. 

The application of the pictorial effects of illumination in 
perspective in general, requires some consideration of the time 
required to make the drawings. The principles developed serve 
as a useful guide, so that the draftsman does not picture impos- 
sible shadows, even though the correct outline is not given. 
In fact, it is a difficult matter exactly to determine the assumed 
direction of the light from the picture itself. For artistic reasons, 
certain portions of an object are purposely subdued in order more 
strongly to emphasize some particular feature. The largest 
application of these principles lies in making illustrations for 
high-class catalogues, particularly for machinery catalogues. Figs. 
272 and 273 give examples of this kind of work. It will bo observed 
that the presented principles are ignored in many respects, yet, 
the effect is pleasing notwithstanding. After all, the theory 
indicates correct modes of procedure, but tlie tiiiu^ re(]uired to 
make such drawings is frequently prohil)itive. lleiu'(\ connnon 
sense, based on mature judgment, nuist be used as a guide. 



288 PICTORIAL EFFECTS OF ILLUMINATION 



QUESTIONS ON CHAPTER XV 

1. When rays of light are parallel, do their perspectives have a common 

vanishing point on the picture plane? Why? 

2. How is the vanishing point of the perspectives of a parallel system 

of hnes found? 

3. What is a visual plane? 

4. Show that the perspective of the intersection of a visual plane and 

the horizontal plane vanishes on the horizon. 

5. If the conventional direction of rays is used, show that the per- 

spectives of their horizontal projections vanish in the right diagonal 
vanishing point. 

6. State the general method of finding the perspective of a shadow 

without first constructing it orthographically. 

7. Show that the method of Question 6 is an application of finding 

the perspectives of intersecting hnes. 

8. A rectangular prism rests on the horizontal plane. Find its shadow 

on that plane by constructing it orthographically and then make 
a perspective of it. 

9. Take the same prism of Question 8 and construct its horizontal 

shadow directly. 

10. An obelisk rests on the horizontal plane. Find its shadow on that 

plane by constructing it orthographically and then make a per- 
spective of it. 

11. Take the same obelisk of Question 10 and construct its horizontal 

shadow directly. 

Note. In the following problems construct the shade and shadow 
orthographically and then find its perspective. 

12. A square-based pyramid rests on a square base. Construct the 

shadow. 

13. A square-based pyramid rests on a circular base. Construct the 

shadow. 

14. A square-based pyramid rests on a hexagonal base. Construct the 

shadow. 

15. A cone rests on a hexagonal base. Construct the shadow. 

16. A rectangular prism rests on two square bases (stepped). Construct 

the shadow. 

17. A rectangular prism rests on two circular bases (stepped). Construct 

the shadow. 

18. A rectangular prism rests on two hexagonal bases (stepped). Con- 

struct the shadow. 

19. A cyhnder has a superimposed square cap. Construct the shadow. 

20. A cyhnder has a superimposed circular cap. Construct the shadow. 

21. A cylinder has a superimposed hexagonal cap. Construct the shadow. 

22. A cyhnder has a superimposed square cap and rests on a square 

base. Construct the shadow. 



n 



IN PERSPECTIVE PROJECTION 289 

23. A cylinder has a superimposed circular cap and rests on a circular 

base. Construct the shadow. 

24. A cyhnder has a superimposed hexagonal cap and rests on a hexagonal 

base. Construct the shadow. 

25. A hollow semi-cyhnder has a superimposed cap. Construct the 

shadows on the inside of the cylinder and on the horizontal plane. 



INDEX 



A ART. NO 

Aerial effect of illumination 1423 

Aerial perspective 1307 

Altitude of a cone 1004 

Altitude of a cylinder 1009 

Angle between curves 921 

Angle between a given line and a given plane 831 

Angle between a given plane and a principal plane 828 

Angle between two given intersecting lines 826 

Angle between two given intersecting planes 827 

Angle, Line drawn through a given point and lying in a given plane 

intersecting the first at a given 836 

Angle of convergence of building lines (perspective) 1331 

Angle of orthographic projection. Advantage of the third 315 

Angle of orthographic projection. First 315 

Angle, Through a given line in a given plane pass another plane intersect- 
ing it at a given 837 

Angle, Visual 1304 

Angles formed by the principal planes 502 

Angles, Isometric projection of 406 

Angles, Lines in all 514 

Angles, Obhque projection of 207 

Angles of orthographic projection 315 

Angles, Points in all 516 

Angles projection to drawing. Application of 317 

Angles, Traces of planes in all 609 

Angles with the i)lanes of projection. Through a given i)oint to draw a 

line of a given length and making given S34 

Angles with the i)rin{'ii)al pianos. Through a given point draw a plane 

making given S35 

Application of angles of projection to drawing 317 

AppHcation of axonometric projection. Comnureial . IIJ 

Application of drawing. Commercial 101 

Application of oblique projection. Commercial 212 

Api)lication orthographic proj(>ction. Conunercial '^1*^ 

Api)licati()n perspective jjrojective. (Commercial !"'•''• 

Approximate metiiod of drawing an ellipse '•*H» 

Archimedian spiral 912 

291 



292 INDEX 

ART. NO. 

Arch, Perspective of 1329 

Arc of a circle 905 

Art of drawing 102 

Asymptote to a hyperbola 908 

Asymptotic surface 1030 

Axes, Angular relation between dimetric 409 

Axes, Angular relation between isometric 402 

Axes, Angular relation between trimetric 410 

Axes, Dimetric 409 

Axes, Direction of dimetric 409 

Axes, Direction of isometric 404 

Axes, Direction of trimetric 410 

Axes, Isometric 402 

Axes, Trimetric 410 

Axis, Conjugate, of a hyperbola 908 

Axis, Major, of an ellipse 906 

Axis, Minor, of an ellipse 906 

Axis of a cone 1004 

Axis of a cylinder 1009 

Axis of a heUx 915 

Axis of a parabola 907 

Axis, Principal, of a hj^erbola (footnote) 908 

Axis, Surfaces of revolution having a common 1025 

Axis, Transverse, of a hyperbola 908 

Axonometric drawing. Commercial application of 412 

Axonometric drawing defined 411 

Axonometric projection. Commercial application of 412 

Axonometric projection defined 411 

B 

Base of a cone 1004 

Base of a cyhnder 1009 

Bell-surface. Intersection of with a plane 1116 

Bounding figures in isometric projection 408 

Bounding figures in oblique projection 210 

Bounding figures in perspective projection 1331 

Branches of a hyperbola 908 

Building, Perspective of 1330 

C 

Centre of a circle 905 

Centre of a curvature 924 

Chord of a circle 905 

Circle defined 905 

Circle, Graduation of the isometric 407 

Circle, Involute of 930 



INDEX 293 

akt.no. 

Circle, Isometric projection of 405 

Circle, Oblique projection of 206 

Circle, Osculating 924 

Circle, Projection of, when it lies in an oblique plane the diameter and 

centre of which are known 838 

Circular cone 1004 

Circular cyhnder 1009 

Classification of hnes 916 

Classification of projections 413, 1332 

Classification of surfaces 1031 

Coincident projections, Lines with 515 

Coincident projections. Points with 517 

Commercial application of axonometric projection 412 

Commercial application of drawing 104 

Commercial application of oblique projection 212 

Commercial application of orthographic projection 318 

Commercial application of perspective projection 1331 

Commercial application of pictorial effects of illumination 1508 

Concave surfaces. Doubly 1022 

Concave surfaces, Line shading applied to 1406 

Concavo-convex surface 1022 

Concentric circles 905 

Concepts, Lines and points considered as mathematical 901, 1001 

Concepts, Mathematical 501, 1001 

Cone, Altitude of 1004 

Cone and sphere. Intersection of 1220 

Cone, Axis of 1004 

Cone, Base of 1104 

Cone, Circular 1004 

Cone defined 1004 

Cone, Development of an intersecting cylinder and 1219 

Cone, Development of an obhque 1119 

Cone, Development when intersected by a plane 1113 

Cone, High-light on 1422 

Cone, Intersection of a cylinder and 1216, 1217, 1218 

Cone, Intersection of a right circular, and a plane 1112 

Cone, Intersection of a sphere and 1220 

Cone of revolution 1004 

Cone, Representation of 1005 

Cone, Right 1004 

Cone, Right circular 1004 

Cone, Slant height of 1001 

Cone, To assume an element on the surface HHH> 

Cone, To assume a point on the surface 1 007 

Conical helix 915 

Conical surfaces 1004 

Conical surfaces, Application of 1113 



294 INDEX 



Conical surfaces, Line of intersection of 1211, 1212, 1213, 1214 

Conical surfaces. Types of line of intersection of 1215 

Conjugate axis of a hyperbola 908 

Contact of tangents. Order of 923 

Convergence of building lines. Angle of 1331 

Convergent projecting lines 1302 

Convex surface. Doubly 1022 

Convex surface. Line shading applied to 1405 

Convolute surface 1013 

Craticulation 1331 

Cube, Perspective of 1317, 1326 

Cube, Shadow of 1414 

Curvature, Centre of 924 

Curvature, Radius of 924 

Curve, Direction of 920 

Curve,Plane 903 

Curved hnes, Doubly 913 

Curved hnes, Line shading applied to 1403 

Curved Unes, Representation of doubly 914 

Curved hnes. Representation of singly 904 

Curved hnes. Singly 903 

Curved surfaces. Development of doubly 1124 

Curved surfaces, Development of doubly (gore method) 1125, 1127 

Curved surfaces. Development of doubly (zone method) 1125 

Curved surfaces. Doubly 1020 

Curved surfaces, Intersection of doubly, with planes 1115 

Curved surfaces of revolution. Doubly 1022 

Curved surfaces of revolution. Intersection of two doublj' 1222 

Curved surfaces of revolution, Representation of doubly 1026 

Curved surfaces of revolution. Singly 1021 

Curved surfaces of revolution, To assume a point on a doubly 1027 

Curved surfaces. Singly 1019 

Curves, Angle between 921 

Curves, Smooth 921 

Cycloid, defined 909 

Cyhnder, defined 1009 

Cylinder, Development of an intersecting cone and 1219 

Cylinder, Development of an oblique 1120 

Cylinder, Development when intersected by a plane 1108 

Cyhnder, High-Ught on 1422 

Cylinder, Intersection of a cone aijd 1216, 1217, 1218 

Cylinder, Intersection of a right circular, and a plane 1107 

Cylinder, Intersection of a, with a sphere 1221 

Cylinder, Representation of 1010 

Cylinder, Shade and shadow on 1417 

Cylinder, To assume an element on the surface 1011 

Cylinder, To assume a point on the surface 1012 



INDEX 295 

AHT. NO. 

Cylinders, Development of two intersecting ..... . 1205, 1207, 1210 

Cylinders, Intersection of two 1204, 1206, 1209 

Cylindrical helix 915 

Cylindrical surfaces 1008 

Cylindrical surfaces, Application of 1109, 1208 



D 

Developable surface 1028, 1 104 

Development by triangulation 1117 

Development of a doubly curved surface by approximation 1124 

Development of a doubly curved surface by the gore method 1127 

Development of a cone intersected by a plane 1113 

Development of a cylinder intersected by a plane 1108 

Development of an intersecting cone and cylinder 1219 

Development of an obhque cone 1119 

Development of an obhque cylinder 1120 

Development of an oblique pyramid 1118 

Development of a prism intersected by a plane 1106 

Development of a pyramid intersected by a plane 1112 

Development of a sphere by the gore method 1121 

Development of a sphere by the zone method 1 126 

Development of a transition piece connecting a circle with a square. . . . 1125 

Development of a transition piece connecting two ellipses 1123 

Development of surfaces 1103 

Development of two intersecting cyHnders 1205, 1207, 1210 

Development of two intersecting prisms 1203 

Diagonal vanishing points. Location of 1325 

Diagonal when applied to perspective 1319 

Diagrams, Transfer of, from orthographic to oblique projection 505, 608 

Diameter of a circle 905 

Dimensions on an orthographic projection 308 

Dimetric axes, Angular relation between 409 

Dimetric drawing 409 

Dimetric projection 409 

Direction of a curve 920 

Direction of light rays. Conventional 1409 

Directrix of a cycloid 909 

Directrix of an epycycloid 910 

Directrix of an hypocycloid 911 

Directrix of a parabola 907 

Directrix of surface 1002, 1003. lOOS 

Directrix, Rectilinear 1003 

Distance between a given plane and a j)lano parrallol to it S29 

Distance between a given point and a given lino S25 

Distance between a given point and a given plane 824 

Distance between two iK)ints in .space 819, 820, 821. 822 



296 INDEX 

ART. NO. 

Distance between two skew lines 832 

Distortion of oblique projection 211 

Drawing an ellipse, Approximate method of 405 

Drawing an ellipse, Exact method of 906 

Drawling, Application of angles of projection to 317 

Drawing, Application of the physical principles of light to 1412 

Drawing, Art of 102 

Drawing, Axonometric 411 

Drawing, Commercial application of 104 

Drawing, Commercial apphcation of axonometric 412 

Drawing, Dimetric 409 

Drawing, Distinction between isometric projection and isometric 403 

Drawing, Examples of isometric 408 

Drawing, Nature of 101 

Drawing of a hne 502 

Drawing, Scales used in making 209 

Drawing, Science of 102 

Drawing to scale 209 

Drawing, Trimetric 410 

Drawing, Use of bounding figures in isometric 408 

Doubl}'^ concave surface 1022 

Doubly convex surface 1022 

Doubly curved line 913 

Doubly curved line, Representation of 914 

Doubly curved surface 1020 

Doubly curved surface. Development by approximation 1124 

Doubly curved surface, Development by the gore method 1127 

Doubly curved surfaces of revolution 1022 

Doubly curved surfaces of revolution, Intersection of ^ 1222 

Doubly curved surfaces of revolution. Intersection of by a plane 1115 

Doubly curved surfaces of revolution. Representation of 1026 

Doubly curved surfaces of revolution, To assume a point on a 1027 

E 

Eccentric circles 905 

Eccentricity of circles 905 

Element, Mathematical 501 

Element of a surface 1002 

Element on the surface of a cone, To assume 1006 

Element on the surface of a cylinder, To assume 1011 

Elevation, defined 301 

EUipse, Approximate method of drawing 405 

EUipse, as isometric projection of a circle 405 

EUipse as obhque projection of a circle 206 

Elhpse defined 906 

Elhpse, Extract method of drawing 906 



INDEX 297 

ART. NO. 

Epycycloid, defined 910 

Evolute, defined 929 

Examples of graduated shades 1426 

F 

Figures in isometric projection, Use of bounding 408 

Figures in oblique projection, Use of bounding 210 

Figures in perspective projection, Use of bounding 1331 

Foci of an ellipse 906 

Foci of an hj^Derbola 908 

Focus of an ellipse 906 

Focus of an hyperbola 908 

Focus of a parabola 907 

Frustiun of a cone 1004 

G 

Generatrix of a surface 1002, 1003, 1008 

Gore method, Development of a doubly curved surface 1127 

Gore method. Development of a sphere 1125 

Graduation shades. Examples of 1426 

Graduation in shade 1424 

Graphic representation of objects 101 

Ground fine, defined 302, 502 

Ground line. Traces of a plane intersecting it 606 

Ground line, Traces of a plane parallel to it 602 

H 

Helix, Conical 915 

Helix defined 915 

Helix, Uniform cylindrical 915 

Height of a cone, Slant 1004 

Helicoid (footnote) 1014 

Hehcoidal screw surface, Oblique 1014 

Hehcoidal screw surface, Right 1015 

High-light 1 US 

High-light on a cylinder or a cone 1 422 

High-light on a sphere 1420 

High-light, Multiple 1421 

Horizon defined 1314 

Horizontal lines inclined to the picture plane, Perspectives of parallel .... 1313 

Horizontal plane, Revolution of in orthographic i)roj(M'tion 303 

Horizontal plane, Revolution of in perspective projection 1322 

Horizontal projection defined 302 

Hyperbola, defined 90S 

Hypocycloid, defined 911 



298 INDEX 



ART. NO. 

Illumination, Aerial effect of 1423 

Illumination, Commercial application of the pictorial effect of 1508 

Incident rays 1419 

Inclined lines, Isometric projection of 406 

Inclined lines. Oblique projection of 207 

Inclined planes. Traces of 605 

Inflexion, Point of 926 

Inflexional tangent 926 

Interpenetration of solids 1215 

Intersecting a given line at a given point, Line 804 

Intersecting in space, Projection of lines 702 

Intersecting lines, Angle between two given 826 

Intersecting line^,, Perspectives of .- 1318 

Intersecting planes, Angle between two given 827 

Intersecting the ground Kne, Traces of planes 606 

Intersection of a beU-sm^face with a plane 1116 

Intersection of a cone and cyhnder 1216, 1217, 1218 

Intersection of a cone and sphere : . . 1220 

Intersection of a doubly cm-ved strrface of revolution and a plane 11 15 

Intersection of a given line at a given point and at a given angle 836 

Intersection of a prism and a plane 1105 

Intersection of a p^^Tamid and a plane 1110 

Intersection of a right circular cone and a plane 1112 

Intersection of a right cii'cular cylinder and a plane 1107 

Intersection of a sphere and a cylinder . 1221 

Intersection of conical sm-faces 1211, 1212, 1213, 1214 

Intersection of conical surfaces. Types of 1215 

Intersection of lines 922 

Intersection of two cylinders 1204, 1206, 1200 

Intersection of two planes obhque to each other and to the principal 

planes 809, 810 

Intersection of two prisms 1202 

Invisible Hues, Representation of 208 

Involute, defined 929 

Involute of a circle 930 

Isometric axes 402 

Isometric axes, Angular relation 402 

Isometric axes, Direction of 404 

Isometric circle, graduation of 407 

Isometric drawing. Distinction between isometric projection and 403 

Isometric drawing, Examples of 408 

Isometric drawing, Use of bounding figures in 408 

Isometric projection and isometric drawing. Distinction between 403 

Isometric projection considered as a special case of orthographic 402 

Isometric projection. Examples of 408 



INDEX 299 



Isometric projection. Nature of 401 

Isometric axes of angles 406 

Isometric projection circles 405 

Isometric projection inclined lines 406 

Isometric projection, Theory of 402 

Isometric projection. Use of bounding figures 408 



Light, Application of the physical principles of, to drawing 1412 

Light, High 1418 

Light, High on a sphere 1420 

Light, Multiple high 1421 

Light, Perspective of parallel rays of 1504 

Light, Physiological effect of 1408 

Light rays, Conventional direction of 1409 

Line, Angle between a given, and a given plane 831 

Lin^, Convergent projecting 1302 

Line, Distance between a given point and a given 825 

Line, Doubly curved 913 

Line, Drawing of 502 

Line fixed in space by its projections 503 

Line, Ground 302, 502 

Line in a plane, pass another plane making a given angle 837 

Line, Indefinite perspective of 1316 

Line intersecting a given fine at a given point 804 

Line intersecting a given fine at a given point and given angle 836 

Line lying in the planes of i)rojection 512 

Line, Meridian 1024 

Line, Oblique plane through a given obhque 806, 807 

Line of a given length making given angles with planes of projection 834 

Line on a given plane. Project a given 830 

Line, Orthographic representation of 502, 504 

Line perpendicular to a given plane through a given point 814, 815 

Line perpendicular to planes of projection 513 

Line, Perspective of 1309, 1321 , 1324 

Line piercing a given plane 823 

Line, Piercing point of, on the principal planes 50() 

Line piercing the principal i)lanes 805 

liine, Plane through a given point, perpendicular to a given 816 

Line, Plane through three given points 817 

Line, Projecting i)lane of 502, 610 

Line, P(>i)ro8entation of doul)ly curved 914 

Line, Uevohition of a point about 707, SIS 

Line, Revolution of a skew 1023 

Line .shading applied to concave surfaces 14(H) 

Lin(> sljuding ai)plie(l to convex surfaces 1405 



300 INDEX 

ART. 1*0. 

Line shading applied to curved lines 1403 

Line shading apphed to plane surfaces 1407 

Line shading applied to sections 1404 

Line .shading apphed to straight hnes 1402 

Line, Singly curved 903 

Line, Singly curved. Representation of 904 

Line, Straight 902 

Lines, Straight, Representation of 904 

Line through a given point parallel to a given line 803 

Line, Traces of a plane intersecting the ground 606 

Line, Traces of planes parallel to the ground 602 

Lines, Angle between two given intersecting 826 

Lines, Classification of 916 

Lines considered as mathematical concepts 901 

Lines, Distance between two skew 832 

Lines in all angles. Projections of 514 

Lines in oblique planes, Projection of 704 

Lines of profile planes. Projection of 518 

Lines intersecting in space. Projection of 702 

Lines, Intersection of 922 

Lines, Isometric projection of inclined 406 

Lines, Line shading apphed to curved 1403 

Lines, Line shading applied to straight 1402 

Lines, Non-intersecting in space, Projection of 703 

Lines, Obhque projection of inclined 207 

Lines, Obhque projection of parallel (footnote) 207 

Lines, Obhque projecting 202 

Lines parallel in space. Projection of 701 

Lines parallel to both principal planes. Perspectives of 1311 

Lines parallel to both principal planes. Projection of 511 

Lines parallel to the plane of projection. Oblique projection of 202 

Lines parallel to the principal planes and lying in an obhque plane .... 705 

Lines perpendicular projecting 302 

Lines perpendicular to given planes, Projection of 706 

Lines perpendicular to the horizontal plane. Perspectives of 1310 

Lines perpendicular to the plane of projection, Obhque projections of . . 204 

Lines perpendicular to the vertical plane, Perspectives of 1312 

Lines, Perspectives of intersecting 1318 

Lines, Perspectives of systems of parallel 1313 

Lines, Representation of invisible 208 

Lines, Representation of visible 208 

Lines, Shadows of 1413 

Lines,' Skew (footnote) 703 

Lines, Systems of, in perspective 1313 

Lines with coincident projections 515 

Linear perspective 1303 

Locus of a generating point 902 



INDEX 301 

M ART. NO. 

Magnitude of objects 103 

Mathematical concepts 501, 901, 1001 

Mathematical elements 501 

Mechanical representation of the principal planes 510 

Meridian line 1024 

Meridian plane 1024 

Multiple high-hght 1421 

N 

Nappes of a conical surface 1003 

Nature of drawing 101 

Nature of isometric projection 401 

Nature of oblique projection 231 

Nature of orthographic projection 301 

Nomenclature of projections 507 

Non-intersecting Hnes in space, Projection of 703 

Normal defined 927 

Normal plane 1018 

O 

Obhque and orthographic projections compared 309 

Oblique cone 1004 

Obhque cylinder 1009 

Oblique helicoidal screw surface 1014 

Obhque hne, Obhque plane through a given 806, 807 

Obhque plane of projection 202 

Oblique plane, Projection of lines in an 704 

Obhque plane, Projection of lines parallel to the principal planes and 

lying in an 705 

Oblique plane through a given oblique line 806, 807 

Oblique plane through a given point 808 

Obhque planes, Intersections of 809, 810 

Oblique projection, Commercial application of 212 

Oblique projection considered as a shadow 203 

Oblique projection, Distortion of 211 

Oblique projection. Examples of 210 

Oblique projection. Location of eye in constructing 202, 211 

Oblique projection. Location of object from plane of projcM'tion 202 

Oblique projection. Nature of 201 

Oblique projection of angles -07 

Oblique projection of circles 20t» 

Obli{}ue i)rojection of inclined lines 207 

Oblique j)rojection of lines i)arallel to the plane of project ion 202 

Obli(jue projection of lines perpendicular to the plane of projection 204 

Oblicjue i)rojecti()n of parallel lines (footnote) 207 



3U2 INDEX 

ART. NO. 

Oblique projection, Theory of 202, 203, 204, 205 

Oblique projection, Transfer of diagrams from orthographic 505, 608 

Oblique projection, Use of bounding figures 210 

Oblique projecting lines 202 

Objects, Graphic representation of 101 

Objects, Magnitude of 103 

Order of contact of tangents 923 

Orthographic and obhque projections compared 309 

Orthographic planes of projection 302 

Orthographic projection, advantages of third angle 315 

Orthographic projection, Angles of 315 

OrthogT-aphic projection considered as a shadow 310 

Orthographic projection. Commercial application of 318 

Orthographic projection, Dimension on 308 

Orthographic projection, First angle of 315 

Orthographic projection. Location of eye in constructing 304, 316 

Orthographic projection. Location of object with respect to the planes 

of projections 308 

Orthographic projection, Nature of 301 

Orthographic projection. Size of object and its projection 305 

Orthographic projection. Theory of 302 

Orthographic projection. Third angle of 315 

Orthographic projection, Transfer of diagrams from oblique 505, 608 

Orthographic projections. Simultaneous interpretation of 308, 318 

Orthogxaphic projections, with respect to each other. Location of 307 

Orthographic representation of hues 502, 504 

Orthographic representation of points ; 508 

Osculating circle 924 

Osculating plane 925 

P 

Pantograph, Use of, in perspective 1331 

Parabola, defined 907 

Parallel hues. Oblique projection of (footnote) 207 

Parallel lines, Orthographic projection of 701 

Parallel Unes to both principal planes, Perspectives of 1311 

Parallel lines to planes of projection, Oblique projection of 202 

Parallel fines to planes of projection, Orthographic projection of 511 

Parallel plane at a given distance from a given jjlane 829 

Parallel to a given plane. Plane which contains a given point and is. . . 813 

Parallel rays of light, Perspectives of 1504 

Parallel systems of lines. Perspectives of 1313 

Parallel to a given fine, Line through a given point 803 

Parallel to the ground fine. Traces of planes 602 

Parallel to the principal planes. Traces of planes 601 

Penumbra 141 1 



INDEX 303 



Perpendicular (when applied to perspective) 1319 

Perpendicular line through a given point to a given plane. 814, 815 

Perpendicular lines to planes of projection, Projection of 513 

Perpendicular lines to the horizontal plane. Perspective of 1310 

Perpendicular plane to a given hne through a given point 816 

Perpendicular projecting lines 302 

Perpendicular to both principal planes. Traces of planes 604 

Perpendicular to given planes. Projection of hues 706 

Perpendicular to one of the principal planes. Traces of planes 603 

Perpendicular to the plane of projection. Oblique projection of hues. . 204 

Perspective, Aerial 1307 

Perspective and shadow of a prism 1502 

Perspective, Commercial application of 1331 

Perspective, Linear 1303 

Perspective of a building 1330 

Perspective of a cube 1317, 1326 

Perspective of a hexagonal prism 1327 

Perspective of a line 1309, 1321, 1324 

Perspective of a line. Indefinite 1316 

Perspective of a line parallel to both principal planes 1311 

Perspective of a line perpendicular to the horizontal plane 1310 

Perspective of a line perpendicular to the vertical plane 1312 

Perspective of a point 1315, 1320, 1323 

Perspective of a pyramid 1328 

Perspective of a shadow. Application of the general method 1506 

Perspective of a shadow, General method of finding 1503 

Perspective of an arch 1329 

Perspective of an obelisk with its shade and shadow 1507 

Perspective of intersecting lines 1318 

Perspective of parallel rays of light 1504 

Perspective of parallel systems of Hues 1313 

Perspective of the horizontal intersection of the visual plane 1505 

Perspective projection, Theory of 1306 

Perspective sketches 1331 

Picture, Center of 1312 

Picture plane 1303, 1308 

Picture plane. Location of 1308 

Pictures 101 

Pictorial effects of illumination. Commercial application of 150S 

Physiological effect of light 1 408 

Plan, defined 301 

Plane, Angle between a given line and a given . 831 

Plane, Anglo between a given i)lane and a i)rincipal 82S 

Plane containing a circle of a known diamoter and (-(MittM-, IVojoction 

of. .• s:^s 

Plane containing a line intersected by another at a givcti point and 

angle .. 836 



30^ INDEX 

ART. NO. 

Plane, Corresponding projection of a given point when in a given. . . 811, 812 

Plane curve 903 

Plane, Distance between a given point and a given 824 

Plane fixed in space by its traces 607 

Plane, Location of picture 1308 

Plane making given angles with the planes of projection 835 

Plane, Meridian 1024 

Plane, Normal 1018 

Plane of projection. Horizontal 302, 502 

Plane of projection, Obhque 202 

Plane of projection, Obhque projection of fines paraUel to 202 

Plane of projection, Obfique projection of lines perpendicular to 204 

Plane of projection. Vertical 302, 502 

Plane, Osculating 925 

Plane paraUel to a given plane at a given distance from it 829 

Plane passed through a fine in a plane making a given angle 837 

Plane, Perpendicular fine through a given point to a given 814, 815 

Plane, Perspective of the horizontal intersection of the visual 1505 

Plane, Picture 1303, 1308 

Plane, Piercing point of a fine on the principal 506, 805 

Plane, Projecting, a given fine on a given 830 

Plane, Projecting, of a line 502, 610 

Plane, Revolution of the horizontal (in orthographic) 303 

Plane, Revolution of the horizontal (in perspective) 1322 

Plane surface 1002 

Plane surfaces. Line shading applied to 1407 

Plane, Tangent 1017 

Plane through a given obfique line, Obfique 806, 807 

Plane through a given point and perpendicular to a given fine 816 

Plane through a given point, Obfique 808 

Plane through three given points 817 

Plane which contains a given point and is parallel to a given plane 813 

Plane, Visual 1310 

Planes, Angles between two given intersecting 827 

Plants, Angles formed by principal 502 

Planes in all angles. Traces of '. 609 

Planes inclined to both principal planes. Traces of 605 

Planes intersecting the ground fine. Traces of 606 

Planes, Intersection of two planes obfique to each other 809, 810 

Planes, Lines in profile 518 

Planes, Line piercing principal 506, 805 

Planes, Mechanical representation of the principal 510 

Planes of projection. Line drawn through a given point, length, and 

angles with 834 

Planes of projection. Lines lying in 512 

Planes of projection. Location of object with respect to 306 

Planes of projection, Orthographic 302, 502 



INDEX 305 

ART. NO. 

Planes of projection, Parallel lines to 511 

Planes of projection, Plane drawn, making given angles with 835 

Planes of projection. Principal 302, 502 

Planes parallel to the principal planes, Traces of 601 

Planes parallel to the ground line, Traces of 602 

Planes perpendicular to both principal planes. Traces of 604 

Planes perpendicular to one of the principal planes, Traces of 603 

Planes, Points lying in the principal, 509 

Planes, Principal '. . 502 

Planes, Profile 311 

Planes, Projection of lines in obUque 704 

Planes, Projection of lines perpendicular to given 706 

Planes, Section 313 

Planes, Supplementary 314 

Point, considered as mathematical concept 901 

Point, Corresponding projection when one is given 811, 812 

Point, Distance between it and a given Une 825 

Point, Distance between it and a given plane 829 

Point, Distance between two, in space 819, 820, 821, 822 

Point, Generating 902 

Point in all angles 516 

Point, Line intersecting another at a given 804 

Point, Line perpendicular to a given plane through a given 814, 815 

Point, Line through a given, parallel to a given line 803 

Point, Location of diagonal vanishing 1325 

Point lying in the principal planes 509 

Point, Oblique plane through a given 808 

Point of inflexion 926 

Point of sight, Choice of 1331 

Point of tangency. To find 919 

Point on a doubly curved surface of revolution. To assume 1027 

Point on a surface of a cone. To assume 1007 

Point on a surface of a cylinder. To assume 1012 

Point, Orthographic representation of 5()S 

Point, Perspective of : 1315. l.TJO, 1323 

Point, Piercing, of a given line on a given piano S23 

Point, Piercing of a given line on the principal pianos 506 

Point, Plane through three given 817 

Point, Plane through a given, perpendicular to a given lino 816 

Point, Plane which contains a given, and is i)arallol to a given piano . . 813 

Point, Revolution of, about a lino 707, SIS 

Point, Through a given, draw a lino of given l(>ngtli and anglo.-^ S34 

Point, Through a given, draw a piano making given angles S35 

Point, Vanishing 1305 

Point with coincident projections 517 

Principal axis of a hyperbola (footnote) 90S 

Principal i)lanes of projection 302. 502 



306 INDEX 

ART. NO. 

Principal planes, Angles formed by 502 

Principal planes, Angle between a given plane and 828 

Principal planes, Intersection of two planes oblique to each other and to 

the 809, 810 

Principal planes. Line piercing 805 

Principal planes, Mechanical representation of 510 

Principal planes. Piercing point of hne on 506 

Principal planes. Points lying in 509 

Principal planes. Traces of planes inclined to both 605 

Principal planes, Traces of planes parallel to 601 

Principal planes. Traces of planes perpendicular to both 604 

Principal planes. Traces of planes perpendicular to one of the 603 

Prism, Developments of two intersecting 1203 

Prism, Developments when intersected by a plane 1106 

Prism, Intersection of two 1202 

Prism, Intersection of with a plane 1105 

Prism, Perspective of 1327 

Prism, Perspective and shadow of 1502 

Prism, Shadow of 1416 

Profile planes 311 

Profile planes. Lines in 518 

Profile projections. Location of 312 

Project a given fine on a given plane 830 

Projecting hne. Convergent 1302 

Projecting hne, Obhque 202 

Projecting line, ParaUel 302, 304 

Projecting plane of a hne 502, 610 

Projection, axonometric 411 

Projection, axonometric, Commercial apphcation of 412 

Projection, Classification of 413, 1332 

Projection, Dimetric 409 

Projection, Isometric, and isometric drawing compared 403 

Projection, Isometric considered as special case of orthographic 402 

Projection, Isometric, Examples of 408 

Projection, Isometric, Nature of 401 

Projection, Isometric of angles 406 

Projection, Isometric of circles 405 

Projection, Isometric of inchned fines 406 

Projection, Isometric, Theory of 402 

Projection, Isometric, Use of bounding figiu-es 408 

Projection, Obhque, Commercial apphcation of 212 

Projection, Oblique considered as shadow 203 

Projection, Obhque, Distortion of 211 

Projection, Oblique, Examples of 210 

Projection, Obhque, Location of object from plane 202 

Projection, Obhque, Nature of 201 

Projection, Obhque of angles 207 



II 



INDEX 307 

ART. NO. 

Projection, Oblique of circles 206 

Projection, Oblique of inclined lines 207 

Projection, Oblique of lines parallel to plane 202 

Projection, Oblique of lines perpendicular to plane 204 

Projection, ObKque of parallel Unes in space (footnote) 207 

Projection, Oblique plane of 202 

Projection, Oblique position of eye in constructing 202, 211 

Projection, ObHque, Theory of 202, 203, 204, 205 

Projection, Oblique, Use of bounding figures 210 

Projection, Orthographic, angles of 315 

Projection, Orthographic, Application of angles 317 

Projection, Orthographic, Commercial application 318 

Projection, Orthographic, Compared with obhque 309 

Projection, Orthographic, Considered as shadow 310 

Projection, Orthographic, Dimensions on 308 

Projection, Orthographic, First angle 315 

Projection, Orthographic, Horizontal plane of 302, 502 

Projection, Orthographic, Line fixed in space by 503 

Projection, Orthographic, Location of object to planes 306 

Projection, Orthographic, Location of observer while constructing. . 304, 316 

Projection, Orthographic, Location of profiles 312 

Projection, Orthographic, Location of projections with respect to each 

other 307 

Projection, Orthographic, Nature of 301 

Projection, Orthographic, Nomenclature of 507 

Projection, Orthographic, of a circle Ij^ing in an oblique plane 838 

Projection, Orthographic, of a point lying in a plane when one iH'ojec- 

tion is given 811, 812 

Projection, Orthographic of lines in oblique planes 704 

Projection, Orthographic of lines intersecting in space 702 

Projection, Orthographic of lines lying in the planes of 512 

Projection, Orthographic of lines non-intersecting in space 703 

Projection, Orthographic of lines parallel in space 701 

Projection, Orthographic of lines parallel to planes of 511 

Projection, Orthographic of lines parallel to one i)hino and in an 

oblique plane 705 

Projection, Orthogra})hic of lines perpendicular to giv(>n planes 706 

Projection, Orthographic of lines perpendicular to planes of 513 

Projection, Orthograpliic of lines with coincident 515 

Projection, Orthograjjliic of points with coincident 517 

Projection, Orthographic, Perpendicular projecting lines 302 

Projection, Orthograi)liic, Plane of 302 

Projection, Ortliographic, Position of eye in constructing 302, 301, 316 

Projection, Orthogra|)hic, Principal planes of 302 

Projection, Orthographic, Size of object and its projection 305 

Projection, Orthographic, Simultaneous interpretation. . 308. 318 

Projection, Ortliographic, Theory of 302 



308 INDEX 

ART. NO. 

Projection, Orthographic, Third angle of projection 315 

Projection, Orthographic, Transfer of diagrams from obHque to. . . . 505, 608 

Projection, Orthographic, Vertical plane of 302, 502 

Projection, Perspective, (see topics under perspective) 

Projection, Scenographic 1302 

Projection, Theory of perspective 1306 

Projection, Trimetric 410 

Pyramid, Development of an obhque 1118 

Pyramid, Development when intersected by a plane 1111 

Pyramid, Intersection of with a plane 1110 

Pyramid, Perspective of 1328 

Pyramid, Shadow of 1415 

Q 

Quadrant of a circle 905 

Radius of a circle 905 

Rays, Conventional direction of light 1409 

Rays, Incident 1419 

Rays of light, Perspective of parallel 1504 

Rays, Reflected '. 1419 

Rays, Visual 1304 

Rectification, defined 928 

Reflected rays 1419 

Representation of cones 1005 

Representation of cyHnders . 1010 

Representation of doubly curved surfaces of revolution 1026 

Representation of invisible Unes 208 

Representation of Unes, Orthographic 502, 504 

Representation of objects, Graphic 101 

Representation of points, Graphic 508 

Representation of singly curved lines 904 

Representation of straight hues 904 

Representation of the principal planes. Mechanical 510 

Representation of visible lines 208 

Revohition, Doubly curved surface of 1022 

Revolution, Cone of 1004 

Revolution, Cylinder of '. 1009 

Revolution, of a point about a line 707, 818 

Revollution of a skew line 1023 

Revo ution of the horizontal plane, (orthographic) 303 

Revolution of the horizontal plane, (perspective) 1322 

Revohition, Representation of doubly curved surfaces of 1026 

Revolution, Singly curved surfaces of 1021 

Revolution, Surfaces of, having a common axis 1025 



INDEX 309 



Revolution, To assume a point on a doubly curved surface of 1027 

Right cone 1004 

Right cylinder 1009 

Right helicoidal screw surface 1015 

Ruled surface 1029 

Rules for shading 1425 

S 

Scale, Choice of 209 

Scale, Drawing to 209 

Scales used in making drawings 209 

Scenographic projection 1302 

Science of drawing 102 

Screw surface, Obhque helicoidal 1014 

Screw surface, Right helicoidal 1015 

Secant of a circle 905 

Section plane 313 

Sections, Line shading applied to 1404 

Sector of a circle 905 

Segment of a circle 905 

Semicircle 905 

Shade, defined 1410 

Shade and shadow of an obelisk in perspective 1507 

Shade and shadow on a cylinder 1417 

Shade, Graduation in 1424 

Shades, Examples of graduated 1426 

Shading apphed to concave surfaces, Line 1406 

Shading applied to convex .surfaces, Line 1405 

Shading applied to curved lines, Line 1403 

Shading ap])lied to plane surfaces. Line 1407 

Shading applied to sections. Line 1404 

Shading applied to straight lines, Line 1402 

Shading rules 1425 

Shadow defined 1410 

Shadow, Application of general method of finding its perspective 1506 

Shadow and perspective of a prism 1502 

Shadow and shade of an obelisk in perspective 1507 

Shadow and shade on a cylindor 1417 

Shadow, General method of finding its perspective 1503 

Shadow, Obliqu(> i)rojection considered as 203 

Shadow of a cube 1414 

Shadow of a line 1413 

Shadow of a prism 1416 

Shadow of a pyramid 1415 

Shadow, Orthograi)hic projection considered iis 310 

Sight, Choice of point of \'<VM 

Singly curved line, defined 903 



310 INDEX 

ART. NO. 

Singly curved line, Representation of 904 

Singly curved surface defined 1019 

Singly curved surface of revolution 1021 

Sketches, Perspective 1331 

Skew lines. Distance between two 832 

Skew lines (footnote) 703 

Skew lines. Revolution of 1023 

Slant height of a cone 1004 

Solids, Interpenetration of 1202, 1215 

Spiral defined 912 

Spiral, Archinicdian 912 

Sphere and cone, Intersection of 1220 

Sphere and cylinder, Intersection of 1221 

Sphere developed by the gore method 1125 

Sphere developed by the zone method 1126 

Sphere, High-light on 1420 

Straight hues defined 902 

Straight hnes, Representation of 904 

Straight lines, Line shading applied to 1402 

Supplementary plane 314 

Surface, Asymptotic 1030 

Surface, Bell, Intersection of, with a plane 1116 

Surface, Concavo- convex 1022 

Surface, Conical 1003 

Surface, Convolute 1013 

Surface, Cylindrical IOCS 

Surface, Developable 1028, 1 103, 1104 

Surface, Doubly concave 1022 

Surface, Doubly convex 1022 

Surface, Doubly curved 1020 

Surface, Doubly curved, Developments by approximation 1124 

Surface, Doubly curved. Developments by the gore method 1127 

Surface, Doubly curved, Intersection of, with a plane 1115 

Surface, Obhque helicoidal screw 1014 

Surface, Plane, defined 1002 

Surface, Right helicoidal screw 1015 

Surface, Ruled 1029 

Surface, Singly curved 1019 

Surface, Warped 1016 

Surface of a cone, To assume a point on the 1007 

Surface of a cone. To assume an element on the 1006 

Surface of a cylinder, To assume a point on the 1012 

Surface of a cylinder, To assume an element on the 1011 

Surface of revolution. Doubly curved 1022 

Surface of revolution, Singly curved 1021 

Surface of revolution. To asusme a point on a doubly curved 1027 

Surfaces, Application of conical 1114, 1211 



INDEX ' 311 

ART. NO. 

Surfaces, Application of cylindrical 1109, 1208 

Surfaces, Classification of 1031 

Surfaces, Development of doubly curved 1124 

Surfaces, Gore method of developing 1124 

Surfaces, Line of intersection of conical 1211, 1212, 1213, 1214 

Surfaces, Line shading applied to concave 1400 

Surfaces, Line shading applied to convex 1405 

Surfaces, Line shading api)lied to plane 1407 

Surfaces of revolution having a common axis 1025 

Surfaces of revolution. Intersection of two doubly 1222 

Surfaces of revolution. Representation of doubly curved 1026 

Surfaces, Types of intersection of conical 1215 

Surfaces, Zone method of developing 1124 

System of lines (in perspective) 1313 



T 

Tangent, defined 917 

Tangent, Construction of 918 

Tangent, Inflexional 926 

Tangent, Order of contact of 923 

Tangent plane 1017 

Tangcnc}^ To find point of 919 

Theory of isometric projection 402 

Theory of oblique projection 202, 203, 204, 205 

Theory of orthographic projection 302 

Theory of pers[)e(;tive j)rojection 130(i 

Theory of shades and shadows 1412 

Torus 1022 

Torus, Development of 1109 

Traces of [)lanes in all angles 609 

Traces of i)lanes inclined to both principal planes 605 

Traces of planes int,erse(;ting the ground line 606 

Tracers of planes i)arallel to the ground line 602 

Traces of i)lan(\s parallel to the i)rincipal planes (iOl 

Traces of i)lanes perpendicular to both principal planes ()01 

Traces of planes perpendicular to one of the princijjal planes 603 

'^Fraces, Plane fixed in space by 607 

Trammel method of drawing an ellipse 90(5 

Transition piece, defined 1121 

"^rransilion i)ioce connc{rting a circle with a square, Development of . . . 1122 

Transition \nrc.v comuHrt ing two elhpses, DevelopnuMit of 1123 

Transv(U'S(; axis of ;i hyjx'rbola iM)S 

Trianguliition, Development by 1117 

Trimetric axes, angular relation 110 

Trimetric drawing 110 

Trimehic projection Ill) 



iN'OVEMBER, 1914 

5H0RT=TITLE CATALOG 

OF THE 

Publications and Importations 

OF 

D. VAN NOSTRAND COMPANY 
25 PARK PLACE, N. Y. 

'Prices marKed tsfith an asterisk (*) are JV£T, 
Alt bifidin^'t are in cloih unless cthertvise noted. 



Abbott, A. V. The Electrical Transmission of Energy 8vo, *$5 oo 

A Treatise on Fuel. (Science Series No. 9.) i6mo, o 53 

Testing Machines. (Science Series No. 74.) i6mo, o 50 

Adam, P. Practical Bookbinding. Trans, by T. E. Maw i2mo, *2 50 

Adams, H. Theory and Practice in Designing 8vo, *2 50 

Adams, H. C. Sewage of Sea Coast Towns 8vo *2 oo 

Adams, J. W. Sewers and Drains for Populous Districts 8vo, 2 50 

Addyman, F. T. Practical X-Ray Work Svo, *4 00 

Adler, A. A. Theory of Engineering Drawing Svo, *2 00 

Principles of Parallel Projecting-line Drawing Svo, *i 00 

Aikman, C. M. Manures and the Principles of Manuring Svo, 2 50 

Aitken, W. Manual of the Telephone Svo, *8 00 

d*Albe, E. E. F., Contemporary Chemistry i2mo, *i 25 

Alexander, J. H. Elementary Electrical Engineering i2mo, 2 00 

Allan, W. Strength of Beams Under Transverse Loads. (Science Series 

No. 19.) i6mo, o 50 

■ Theory of Arches. (Science Series No. 11.) i6mo, 

Allen, H. Modern Power Gas Producer Practice and Applications. lamo, *2 50 

Gas and Oil Engines Svo, *4 50 

Anderson, F. A. Boiler Feed Water Svo, *2 5a 

Anderson, Capt. G. L. Handbook for the Use of Electricians Svo, 3 00 

Anderson, J. W. Prospector's Handbook i2mo, i 50 

And6s, L. Vegetable Fats and Oils Svo, *4 00 

Animal Fats and Oils. Trans, by C. Salter Svo, *4 00 

Drying Oils, Boiled Oil, and Solid and Liquid Driers Svo, *5 00 

Iron Corrosion, Anti-fouling and Anti-corrosive Paints. Trans, by 

C. Salter Svo, *4 00 

And^s, L. Oil Colors, and Printers' Ink. Trans, by A. Morris and 

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i2mo, *2 50 



D. VAN NOSTRAXD CO.'S SHORT TITLE CATALOG 3 

Andrews, E. S. Reinforced Concrete Construction i2nio, "^i 25 

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Andrews, E. S., and Heywood, H. B. The Calculus for Engineers. larao, *i 25 
Annual Reports on the Progress of Chemistry. Nine Volumes now ready. 

Vol. I. 1904, Vol. IX, 1912 8vo, each, 2 00 

Argand, M. Imaginary Quantities. Translated from the French by 

A. S. Hardy. (Science Series No. 52.) i6mo, o 50 

Armstrong, R., and Idell, F. E. Chimneys for Furnaces and Steam Boilers. 

(Science Series No. i.) i6mo, o 50 

Arnold, E. Armature Windings of Direct-Current Dynamos. Trans, by 

F. B. DeGress 8 vo, *2 00 

Asch, W., and Asch, D. The Silicates in Chemistry and Commerce. 8vo, *6 00 
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Ashe, S. W. Electric Railways. Vol. II. Engineering Preliminaries and 

Direct Current Sub-Stations i2mo, 

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Ashley, R. H. Chemical Calculations {In Press.) 

Atkins, W. Common Battery Telephony Simplified i2mo, 

Atkinson, A. A. Electrical and Magnetic Calculations Svo, 

Atkinson, J. J. Friction of Air in Mines. (Science Series No. 14.) . . i6mo, 
Atkinson, J. J., and Williams, Jr.*, E. H. Gases Met with in Coal Mines. 

(Science Series No. 13.) i6mo, 

Atkinson, P. The Elements of Electric Lighting i2mo, 

The Elements of Dynamic Electricity and Magnetism i2mo, 

Power Transmitted by Electricity i2mo, 

Auchincloss, W. S. Link and Valve Motions Simplified Svo, 

Austin, E. Single Phase Electric Railways 4to, 

Ayrton, H. The Electric Arc Svo, 

Bacon, F. W. Treatise on the Richards Steam-Engine Indicator . . i2mo, 

Bailes, G. M. Modern Mining Practice. Five Volumes Svo, each, 

Bailey, R. D. The Brewers' Analyst Svo, 

Baker, A. L. Quaternions Svo, 

■ Thick-Lens Optics i2mo, 

Baker, Benj. Pressure of Earthwork. (Science Series No. 56.)...i6mo, 

Baker, I. O. Levelling. (Science Series No. 91.) i6mo, 

Baker, M. N. Potable Witer. (Science Series No. 61.) i6mo, 

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Baker, T. T. Telegraphic Transmission of Photographs r2mc, 

Bale, G. R. Modern Iron Foundry Practice. Two Volumes. i2mo. 

Vol. I. Foundry Equipment, Materials Used 

Vol. II. Machine Moulding and Moulding Machines 

Bale, M. P. Pumps and Pumping. 1 2mo, 

Ball, J, W. Concrete Structures in Railways Svo, 

Ball, R. S. Popular Guide to the Heavens „ Svo, 

Natural Sources of Power. (Westminster Series.) Svo, 

Ball, W. V. Law Affecting Engineers Svo, 

Bankson, Lloyd. Slide Valve Diagrams. (Science Series No. 108.) . i6mo, 



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4 D. VAX NOSTRAND CO.'S SHORT TITLE CATALOG 

Barba, J. Use of Steel for Constructive Purposes i2mo, i oo 

Barham, G. B, Development of the Incandescent Electric Lamp. . . .8vo, *2 oo 

Barker, A. F. Textiles and Their Manufacture. (Westminster Series.) 8 vo, 2 00 

Barker, A, F., and Midgley, E. Analysis of Textile Fabrics 8vo, 3 00 

Barker, A. H. Graphic Methods of Engine Design i2mo, *i 50 

" Heating and Ventilaiion 4to, *8 00 

Barnard, J. H. The Naval Militiaman's Guide i6mo, leather i 00 

Barnard, Major J. G. Rotary Motion. (Science Series No. 90.). .. . i6mo, o 50 

Barrus, G. H. Boiler Tests 8vo, *3 00 

Engine Tests Svo, *4 00 

The above two purchased together *6 00 

Barwise, S. The Purification of Sewage i2mo, 3 50 

Baterden, J. R. Timber. (Westminster Series.) 8^0, -^2 oo 

Bates, E. L., and Charlesworth, F, Practical Mathematics i2mo, 

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Part n. Advanced Course . . *i 50 

■ Practical Mathematics i2mo, *i 50 

Practical Geometry and Graphics i2mo, *2 oo 

Batey, J. The Science of Works Management lamo, *i 25 

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Beaumont, R. Color in Woven Design 8vo, *6 00 

Finishing of Textile Fabrics Svo, *4 co 

Beaumont, W. W. The Steam-Engine Indicator Svo, 2 50 

Bechhold, H. Colloids in Biology and Medicine. Trans, by J. G. 

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Beckwith, A. Pottery Svo, paper, o 60 

Bedell, F., and Pierce, C. A. Direct and Alternating Current Manual. 

Svo, *2 00 

Beech, F. Dyeing of Cotton Fabrics Svo, *3 00 

. Dyeing of Woolen Fabrics Svo, *3 50 

Begtrup, J. The Slide Valve Svo, *2 o3 

Beggs, G. E. Stresses in Railway Girders and Bridges {In Press.) 

Bender, C. E. Continuous Bridges. (Science Series No. 26.).. .... i6mo, o 50 

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Bernthsen, A. A Text - book of Organic Chemistry. Trans, by G. 

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Wright Svo, *5 00 

Bertin, L. E. Marine Boilers. Trans, by L. S. Robertson Svo, 5 00 

Beveridge, J. Papermaker's Pocket Book i2mo, *4 00 

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Binns, C. F. Ceramic Technology Svo, *5 00 

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Blaine, R. G. The Calculus and Its Applications . . i2mo, *i 50 

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Blasdale, W. C. Quantitative Chemical Analysis. .. .lamo, (In Press.) 

Bligh, W. G. The Practical Design of Irrigation Works 8vo, *6 00 

Bloch, L. Science of Illumination. Trans, by W. C. Clinton S'vo, *2 50 

Blok, A. Illumination and Artificial Lighting lamo, i 25 

Bliicher, H. Modem Industrial Chemistry. Trans, by J. P. Millington. 

bvo, *7 50 

Blyth, A. W. Foods: Their Composition and Analysis 8vo, 7 50 

— — Poisons: Their Effects and Detection bvo, 7 50 

Bockmann, F. Celluloid i2mo, *2 50 

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Superheaters and Superheating and Their Control Bvo, *i 50 

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Bottone, S. R. Magnetos for Automobilists i2mo, *i 00 

Boulton, S. B. Preservation of Timber. (Science Series No. B2.) . i6mo, o 50 

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Bow, R. H. A Treatise on Bracing Bvo, i 50 

Bowie, A, J., Jr. A Practical Treatise on Hydraulic Mining Bvo, 5 00 

Bowker, W. R. Dynamo, Motor and Switchboard Circuits Bvo, *2 50 

Bowles, O. Tables of Common Rocks. (Science Series No. i25.).i6mo, o 50 

Bowser, E. A. Elementary Treatise on Analytic Geometry i2mo, i 75 

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Elementary Treatise on Hydro-mechanics i2mo, 2 50 

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Bragg, E. M. Marine Engine Design i2mo, *2 00 

Design of Marine Engines and Auxiliaries (In Press.) 

Brainard, F. R. The Sextant. (Science Series No. loi.) i6mo, 

Brassey's Naval Annual for 191 1 Bvo, *6 00 

Brew, W. Three-Phase Transmission Bvo, *2 00 

Briggs, R., and Wolff, A. R. Steam-Heating. (Science Series No. 

67.) i6mo, o 50 

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Brislee, T. J. Introduction to the Study of Fuel. (OutUnes of Indus- 
trial Chemistry. ) 8vo, *3 00 

Broadfoot, S. K. Motors, Secondary Batteries. (Installation Manuals 

Series. ) 1 2rao, *o 75 

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Brown, G. Healthy Foundations. (Science Series No. 80.) i6mo, o 50 



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Brown, H. Irrigation 8vo, *5 oo 

Brown, Wm. N. The Art of Enamelling on Metal i2mo, *i oo 

Handbook on Japanning and Enamelling i2mo, *i 50 

■ House Decorating and Painting i2mo, *i 50 

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Brown, Wm, N. Dipping, Burnishing, Lacquering and Bronzing 

Brass Ware i2mo, *i 00 

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Browne, C. L. Fitting and Erecting of Engines 8vo, *i 50 

Browne, R. E. Water Meters. (Science Series No. 81.) i6mo, o 50 

Bruce, E. M. Pure Food Tests i2mo, *i 25 

Bruhns, Dr. New Manual of Logarithms 8vo, cloth, 2 00 

half morocco, 2 50 
Brunner, R. Manufacture of Lubricants, Shoe Polishes and Leather 

Dressings. Trans, by C. Salter 8vo, *3 00 

Buel, R. H. Safety Valves. (Science Series No. 21.) i6mo, o 50 

Burns, D. Safety in Coal Mines i2mo, *i 00 

Burstall, F. W. Energy Diagram for Gas. With Text 8vo, i 50 

Diagram. Sold separately *i 00 

Burt, W. A. Key to the Solar Compass ^ i6mo, leather, 2 50 

Burton, F. G. Engineering Estimates and Cost Accounts i2mo, *i 50 

Buskett, E. W. Fire Assaying • i2mo, *i 25 

B utler, H. J. Motor Bodies and Chassis 8vo, *2 50 

Byers, H. G., and Knight, H. G. Notes on Qualitative Analysis . . . .8vo, *i 50 

Cain, W. Brief Course in the Calculus i2mo, *i 75 

Elastic Arches. (Science Series No. 48.) i6mo, o 50 

' Maximum Stresses. (Science Series No. 38.) i6mo, o 50 

— — Practical Designing Retaining of Walls. (Science Series No. 3.) 

i6mo, o 50 

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(Science Series No. 42.) i6mo, o 50 

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Symbolic Algebra. (Science Series No. 73.) i6mo, o 50 

Campin, F. The Construction of Iron Roofs 8vo, 2 00 

Carpenter, F. D. Geographical Surveying. (Science Series No. 37.).i6mo, 

Carpenter, R. C, and Diederichs, H. Internal Combustion Engines. . 8vo, *5 00 

Carter, E. T. Motive Power and Gearing for Electrical Machinery . 8vo, *5 00 

Carter, H. A. Ramie (Rhea), China Grass i2mo, *2 00 

Carter, H. R. Modern Flax, Hemp, and Jute Spinning 8vo, *3 00 

Cary, E. R. Solution of Railroad Problems with the Slide Rule. . i6mo, *i 00 

Cathcart, W. L. Machine Design. Part I. Fastenings 8vo, *3 00 

Cathcart, W. L., and Chaffee, J. I. Elements of Graphic Statics . . .8vo, *3 00 

— — Short Course in Graphics i2mo, i 50 

Caven, R. M., and Lander, G. D. Systematic Inorganic Chemistry. i2mo, *2 00 

Chalkley, A. P. Diesel Engines 8vo, *3 00 

Chambers' Mathematical Tables 8vo, i 75 

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Charpentier, P. Timber 8vo, *6 00 






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D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

Chatley, H. Principles and Designs of Aeroplanes. (Science Series 

No. 126) i6mo, 

How to Use Water Power i2mo, 

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Salter i2mo, 

Christie, W. W. Boiler-waters, Scale, Corrosion, Foaming 8vo, 

Chimney Design and Theory 8vo, 

Furnace Draft. (Science Series No. 123.) i6mo, 

Water: Its Purification and Use in the Industries Svo, 

Church's Laboratory Guide. Rewritten by Edward Kinch Svo, 

Clapperton, G. Practical Papermaking Svo, 

Clark, A. G. Motor Car Engineering. 

Vol. I. Construction *3 00 

Vol. II. Design {In Press.) 

Clark, C. H. Marine Gas Engines i2mo, *i 50 

Clark, D. K. Fuel: Its Combustion and Economy i2mo, i 50 

Clark, J. M. New System of Laying Out Railway Turnouts i2mo, i 00 

Clarke, J. W., and Scott, W. Plumbing Practice. 

Vol. I. Lead Working and Plumbers' Materials Svo, *4 00 

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Clausen-Thue, W. ABC Telegraphic Code. Fourth Edition . . . i2mo, *5 00 
Fifth Edition Svo, *7 00 

The A I Telegraphic Code Svo, *7 50 

Clerk, D., and Idell, F. E. Theory of the Gas Engine. (Science Series 

No. 62.) i6mo, o 50 

Clevenger, S. R. Treatise on the Method of Government Surveying. 

i6mo, morocco, 

Clouth, F. Rubber, Gutta-Percha, and Balata Svo, 

Cochran, J. Concrete and Reinforced Concrete Specifications Svo, 

Treatise on Cement Specifications Svo, 

Coffin, J. H. C. Navigation and Nautical Astronomy i2mo, 

Colburn, Z., and Thurston, R. H. Steam Boiler Explosions. (Science 

Series No. 2.) i6mo, 

Cole, R. S. Treatise on Photographic Optics i2mo, 

Coles-Finch, W. Water, Its Origin and Use Svo, 

Collins, J. E. Useful Alloys and Memoranda for Goldsmiths, Jewelers. 

i6mo, 

Collis, A. G. High and Low Tension Switch-Gear Design Svo, 

Switchgear. (Installation Manuals Series. ) i2mo, 

Constantine, E. Marine Engineers, Their Qualifications and Duties. Svo, 

Coombs, H. A. Gear Teeth. (Science Series No. 120.) i6mo. 

Cooper, W. R. Primary Batteries Svo, 

" The Electrician " Primers Svo, 

Part I 

Part II 

Part III 



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8 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

Copperthwaite, W. C. Tunnel Shields 4to, *g 00 

Corey, H. T. Water Supply Engineering 8vo {In Press.) 

Corfield, W. H. Dwelling Houses. (Science Series No. 50.) .... i6mo, 

Water and Water-Supply. (Science Series No. 17.) i6mo, 

Cornwall, H. B. Manual of Blow-pipe Analysis 8vo, 

Courtney, C. F. Masonry Dams 8vo, 

Cowell, W. B. Pure Air, Ozone, and Water i2mo, 

Craig, T. Motion of a Solid in a Fuel. (Science Series No. 49.) . i6mo, 

Wave and Vortex Motion. (Science Series No. 43.) i6mo, 

Cramp, W. Continuous Current Machine Design 8vo, 

Creedy, F. Single Phase Commutator Motors 8vo, 

Crocker, F. B. Electric Lighting. Two Volumes. 8vo. 

Vol. I. The Generating Plant 303 

Vol. II. Distributing Systems and Lamps 

Crocker, F. B., and Arendt, M. Electric Motors 8vo, *2 50 

Crocker, F. B., and Wheeler, S. S. The Management of Electrical Ma- 
chinery i2mo, *i 00 

Cross, C. F., Bevan, E. J., and Sindall, R. W. Wood Pulp and Its Applica- 
tions. (Westminster Series.) Svo, 

Crosskey, L. R. Elementary Perspective 8vo, 

Crosskey, L. R., and Thaw, J. Advanced Perspective Svo, 

CuUey, J. L. Theory of Arches. (Science Series No. 87.) i6mo, 

Dadourian, H. M. Analytical Mechanics i2mo, 

Danby, A. Natural Rock Asphalts and Bitumens 8vo, 

Davenport, C. The Book. (Westminster Series.) Svo, 

Davey, N. The Gas Turbine Svo, 

Davies, D. C. Metalliferous Minerals and Mining Svo, 

Earthy Minerals and Mining Svo, 

Davies, E. H. Machinery for Metalliferous Mines Svo, 8 00 

Davies, F. H. Electric Power and Traction . Svo, *2 00 

Foundations and Machinery Fixing. (Installation Manual Series.) 

i6mo, 

Dawson, P. Electric Traction on Railways Svo, 

Day, C. The Indicator and Its Diagrams i2mo, 

Deerr, N. Sugar and the Sugar Cane Svo, 

Deite, C. Manual of Soapmaking. Trans, by S. T. King 4to, 

DelaCoux, H. The Industrial Uses of Water. Trans, by A. Morris. Svo, 

Del Mar, W. A. Electric Power Conductors Svo, 

Denny, G. A. Deep-level Mines of the Rand 4to, 

Diamond Drilling for Gold 

De Roos, J. D. C. Linkages. (Science Series No. 47.) i6mo, 

Derr, W. L. Block Signal Operation Oblong i2mo, 

— ^ Maintenance-of-Way Engineering (In Preparation.) 

Desaint, A. Three Hundred Shades and How to Mix Them Svo, 

De Varona, A. Sewer Gases. (Science Series No. 55.) i6mo, 

Devey, R. G. Mill and Factory Wiring. (Installation Manuals Series.) 

i2mo, 

Dibdin, W. J. Public Lighting by Gas and Electricity Svo, 

Purification of Sewage and Water Svo, 



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D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 9 

Eichmann, Carl. Basic Open-Hearth Steel Process i2mo, *3 50 

Dieterich, K. Analysis of Resins, Balsams, and Gum Resins 8vo, *3 00 

Dinger, Lieut. H. C. Care and Operation of Naval Machinery . . . i2mo, *2 00 
Dixon, D. B. Machinist's and Steam Engineer's Practical Calculator. 

i6mo, morocco, i 25 
Doble, W. A. Power Plant Construction on the Pacific Coast {In Press.) 

Dommett, W. E. Motor Car Mechanism lamo, *i 25 

Dorr, B. F. The Surveyor's Guide and Pocket Table-book. 

i6mo, morocco, 

Down, P. B. Handy Copper Wire Table i6mo. 

Draper, C. H. Elementary Text-book of Light, Heat and Sound . . i2mo, 

Heat and the Principles of Thermo-dynamics i2mo, 

Dubbel, H. High Power Gag Engines 8vo, 

Duckwall, E. W. Canning and Preserving of Food Products 8vo, 

Dtunesny, P., and Noyer, J. Wood Products, Distillates, and Extracts. 

8vo, *4 50 
Duncan, W. G., and Penman, D. Th© Electrical Equipment of Collieries. 

8vo, *4 00 
Dunstan, A. E., and Thole, F. B. T. Textbook of Practical Chemistry. 

i2mo, *i 40 
Duthie, A. L. Decorative Glass Processes. (Westminster Series.) . Bvo, *2 00 

Dwight, H. B. Transmission Line Formulas Bvo, *2 00 

Dyson, S. S. Practical Testing of Raw Materials 8vo, *5 00 

Dyson, S. 8., and Clarkson, S. S. Chemical Works Bvo, *7 50 

Eccles, R. G., and Duckwall, E. W. Food Preservatives . . . Bvo, paper, o 50 
Eck, J. Light, Radiation and Illumination. Trans, by Paul Hogner, 

8vo, 

Eddy, H. T. Maximum Stresses under Concentrated Loads 8vo, 

Edelman, P. Inventions and Patents i2mo. {In Press.) 

Edgcumbe, K. Industrial Electrical Measuring Instruments Bvo, 

Edler, R. Switches and Switchgear. Trans, by Ph. Laubach. . Bvo, 

Eissler, M. The Metallurgy of Gold Bvo, 

— — The Hydrometallurgy of Copper Bvo, 

The Metallurgy of Silver Bvo, 

— — The Metallurgy of Argentiferous Lead Bvo, 

A Handbook on Modern Explosives Bvo, 

Ekin, T. C. Water Pipe and Sewage Discharge Diagrams folio, 

Eliot, C. W., and Storer, F. H. Compendious Manual of Qualitative 

Chemical Analysis i2mo, 

Ellis, C. Hydrogenation of Oils Bvo, 

Ellis, G. Modern Technical Drawing Bvo, 

Ennis, Wm. D. Linseed Oil and Other Seed Oils Bvo, 

• Applied Thermodynamics Bvo, 

Flying Machines To-day 1 2mo, 

Vapors for Heat Engines 1 2mo, 

Erfurt, J. Dyeing of Paper Pulp. Trans, by J. Hubner Bvo, 

Ermen, W. F. A. Materials Used in Sizing Bvo, 

Evans, C. A. Macadamized Roads {Inrirs<.) 



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10 D. \AX XOSTRAXD CO.'S SHORT TITLE CAT.vLOG 

Ewing, A. J. Magnetic Induction in Iron 8vo, *4 oc 

Fairie, J. Notes on Lead Ores i2mo, ^ 

Notes on Pottery Clays i2mo, 

Fairley, W., and Andre, Geo. J. Ventilation of Coal Mines. (Science 

Series No. 58.) i6mo, 

Fair weather, W. C. Foreign and Colonial Patent Laws 8vo, 

Fanning, J. T. Hydraulic and Water-supply Engineering 8vo, 

Fauth, P. The Moon in Modem Astronomy. Trans, by J. McCabe. 

8vo, 

Fay, I. W. The Coal-tar Colors 8vo, 

Fembach, R. L. Glue and Gelatine 8vo, 

Chemical Aspects of Silk Manufacture i2mo, 

Fischer, E. The Preparation of Organic Compounds. Trans, by R. V. 

Stanford i2mo, 

Fish, J. C. L. Lettering of Working Drawings Oblong 8vo, 

Fisher, H. K. C, and Darby, W. C. Submarine Cable Testing . . . .8vo, 
Fleischmann, W. The Book of the Dairy. Trans, by C. M. Aikman. 

8vo, 4 00 
Fleming, J. A. The Alternate-current Transformer. Two Volumes. 8vo. 

Vol. I. The Induction of Electric Currents *5 00 

Vol. n. The Utilization of Induced Currents *5 00 

Fleming, J. A. Propagation of Electric Currents 8vo, *3 00 

Centenary of the Electrical Current 8vo, *o 50 

— - — Electric Lamps and Electric Lighting 8vo, *3 00 

Electrical Laboratory Notes and Forms 4to, *5 00 

A Handbook for the Electrical Laboratory and Testing Room. Two 

Volumes 8vo, each, *5 oo 

Fleury, P. Preparation and Uses of White Zinc Paints 8vo, *2 50 

Fleury, H. The Calculus Without Limits or Infinitesimals. Trans, by 

C. O. Mailloux ' //. Press.) 

Flynn, P. J. Flow of Water. (Science Series No. 84.) i2mo, o 50 

HydrauUc Tables. (Science Series No. 66.) i6mo, o 50 

Foley, N. British and American Customary and Metric Measures . .folio, *3 oo 
Forgie, J. Shield Tunneling 8vo. (/;; Press.) 

Foster, H. A. Electrical Engineers' Pocket-book. (Seventh Edition.) 

i2mo, leather, s 00 

Engineering Valuation of Public UtiUties and Factories 8vo, *3 00 

Handbook of Electrical Cost Data 8vo In Press.) 

Foster, Gen. J. G. Submarine Blasting in Boston (Mass.) Harbor 4to, 3 50 

Fowle, F. F. Overhead Transmission Line Crossings i2mo, *i 50 

The Solution of Alternating Current Problems 8vo [In Press.) 

Fox, W. G. Transition Curves. (Science Series No. no.) i6mo, 50 

Fox, W., and Thomas, C. W. Practical Course in Mechanical Draw- 
ing i2mo, 

Foye, J. C. Chemical Problems. (Science Series No. 69. "> i6mo, 

Handbook of Mineralogy. (Science Series No. 86.) i6mo, 

Francis, J. B. Lowell HydrauUc Experiments 4to, 

Franzen, H. Exercises in Gas Analysis i2mo, 



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50 


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D. VAN KOSTRAND CO.'S SHORT TITLE CATALOG n 

Freudemacher, P. W. Electrical Mining Installations. (Installation 

Manuals Series.) i2mo, 

Frith, J. Alternating Current Design 8vo, 

Fritsch, J. Manufacture of Chemical Manures. Trans, by D. Grant. 

8vo, 
Frye, A. I. Civil Engineers' Pocket-book i2mo, leather, 

Fuller, G. W. Investigations into the Purification of the Ohio River. 

4to, 
Fumell, J. Paints, Colors, Oils, and Varnishes 8vo. 

Gairdner, J. W. I. Earthwork 8vo {hi Press.) 

Gant, L. W. Elements of Electric Traction 8vo, 

Garcia, A. J. R. V. Spanish-English Railway Terms 8vo, 

Garforth, W. E. Rules for Recovering Coal Mines after Explosions and 

Fires i2mo, leather, 

Gaudard, J. Foundations. (Science Series No. 34.) i6mo, 

Gear, H. B., and Williams, P. F. Electric Central Station Distribution 

Systems 8vo, 

Geerligs, H. C. P. Cane Sugar and Its Manufacture 8vo, 

World's Cane Sugar Industry 8vo, 

Geikie, J. Structural and Field Geology 8vo, 

Mountains. Their Growth, Origin and Decay 8vo, 

The Antiquity of .Man in Europe 8vo, 

Georgi, P., and Schubert, A. Sheet Metal Working. Trans, by C. 

Salter 8vo, 

Gerber, N. Analysis of Milk, Condensed Milk, and Infants' Mjlk-Food. Svo, 
Gerhard, W. P. Sanitation, Watersupply and Sewage Disposal of Country 

Houses i2mo, 

Gas Lighting (Science Series No. iii.) i6mo, 

Household Wastes. (Science Series No. 97.) i6mo, 

House Drainage. (Science Series No. 63.) i6mo, 

Gerhard, W. P. Sanitary Drainage of Buildings. (Science Series No. 93.) 

i6mo, 

Gerhardi, C. W. H. Electricity Meters 8vo, 

Geschwind, L. Manufacture of Alum and Sulphates. Trans, by C. 

Salter Svo, 

Gibbs, W. E. Lighting by Acetylene i2mo, 

Physics of Solids and Fluids. (Carnegie Technical School's Text- 
books.) *i 

Gibson, A. H. Hydraulics and Its Application Svo, 

Water Hammer in Hydraulic Pipe Lines i2mo, 

Gilbreth, F. B. Motion Study i2mo, 

Primer of Scientific Management i2mo, 

Gillmore, Gen. Q. A. Limes, Hydraulic Cements acd Mortars Svo, 

Roads, Streets, and Pavements i2rao, 

Golding, H. A. The Theta-Phi Diagram i2mo, 

Goldschmidt, R. Alternating Current Commutator Motor Svo, 

Goodchild, W. Precious Stones. (Westminster Series.) Svo, 

Goodeve, T. M. Textbook on the Steam-engine i2mo, 

Gore, G. Electrolytic Separation of Metals Svo, 



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12 D. VAX XOSTRAXD CO.'S SHORT TITLE CATALOG 

Gould, E. S. Arithmetic of the Steam-engine i2mo, 

Calculus. (Science Series No. 112.) i6mo, 

High Masonry Dams. (Science Series No. 22.) i6mo, 

Practical Hydrostatics and Hydrostatic Formulas. (Science Series 

No. 117.; i6mo, 

Gratacap, L. P. A Popular Guide to Minerals 8vo, 

Gray, J. Electrical Influence Machines i2mo, 

Marine Boiler Design i2mo, 

Greenhill, G. Dynamics of Mechanical Flight 8vo, 

Greenwood, E. Classified Guide to Technical and Commercial Books. 8vo, 

Gregorius, R. Mineral Waxes. Trans, by C. Salter. i2mo, 

Griffiths, A. B. A Treatise on Manures i2mo, 

Dental Metallurgy 8vo, 

Gross, E. Hops 8vo, 

Grossman, J. Ammonia and Its Compounds i2mo, 

Groth, L. A. Welding and Cutting Metals by Gases or Electricity. 

I Westminster Series 1 8vo, 

Grover, F. Modern Gas and Oil Engines 8vo, 

Gruner, A. Power-loom Weaving 8vo, 

Giildner, Hugo. Internal Combustion Engines. Trans, by H. Diederichs. 

4to, 

Gunther, C. 0. Integration i2mo. 

Garden, R. L. Traverse Tables folio, half morocco, 

Guy, A, E. Experiments on the Flexure of Beams 8vo, 

Haeder, H. Handbook on the Steam-engine. Trans, by H. H. P. 

Powles * i2mo, 

Hainbach, R. Pottery Decoration. Trans, by C. Salter i2mo, 

Haenig, A. Emery and Emery Industry 8vo, 

Hale, W. J. Calculations of General Chemistry i2mo. 

Hall, C. H. Chemistry of Paints and Paint Vehicles i2mo, 

Hall, G. L. Elementary Theory of Alternate Current Working. .. .8vo, 

Hall, R. H. Governors and Governing Mechanism i2mo. 

Hall, W. S. Elements of the Differential and Integral Calculus 8vo, 

Descriptive Geometry 8vo volume and a 4to atlas, 

Haller, G. F., and Cunningham, E. T. The Tesla Coil i2mo, 

Halsey, F. A. Slide Valve Gears i2mo, 

■ The Use of the Slide Rule. (Science Series No. 114.) i6mo, 

Worm and Spiral Gearing. (Science Series No. 116.) i6mo, 

Hamilton, W. G. Useful Information for Railway Men i6mo, 

Hammer, W. J. Radium and Other Radio-active Substances 8vo, 

Hancock, H. Textbook of Mechanics and Hydrostatics 8vo, 

Hancock, W. C. Refractory Materials. ( Metallurgy Series.; (/;; Press.) 

Hardy, E. Elementary Principles of Graphic Statics i2mo, *i 50 

Harris, S. M. Practical Topographical Surveying {In Press.) 

Harrison, W. B. The Mechanics' Tool-book i2mo, i 50 

Hart, J. W. External Plumbing Work 8vo, *3 00 

Hints to Plumbers on Joint Wiping 8vo, *3 00 

Principles of Hot Water Supply 8vo. *3 00 

Sanitary Plumbing and Drainage 8vo, *3 00 



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D. VAN NOSTRAND CO.'S SHORT TITLE CATaLOG 13 

Haskins, C. H. The Galvanometer and Its Uses i6mo, i 50 

Hatt, J. A. H. The Colorist square i2mo, *i 50 

Hausbrand, E. Drying by Means of Air and Steam. Trans, by A. C. 

Wright i2mo, *2 00 

Evaporating, Condensing and Cooling Apparatus. Trans, by A. C. 

Wright 8vo, *5 00 

Hausner, A. Manufacture of Preserved Foods and Sweetmeats. Trans. 

by A. Morris and H. Robson 8vo, 

Hawke, W. H. Premier Cipher Telegraphic Code 4to, 

100,000 Words Supplement to the Premier Cocie 4to, 

Hawkesworth, J. Graphical Handbook for Reinforced Concrete Design. 

4to, 

Hay, A. Alternating Currents 8vo, 

Electrical Distributing Networks and Distributing Lines. Svo, 

Continuous Current Engineering 8vo, 

Hayes, H. V. Public Utilities, Their Cost New and Depreciation. . .8vo, 

Heap, Major D. P. Electrical Appliances Svo, 

Heather, H. J. S. Electrical Engineering Svo, 

Heaviside, O. Electromagnetic Theory. Vols. I and II. . . Svo, each. 

Vol. Ill Svo, 

Heck, R. C. H. The Steam Engine and Turbine Svo, 

Steam-Engine and Other Steam Motors. Two Volumes. 

Vol. I. Thermodynamics and the Mechanics Svo, 

Vol. II. Form, Construction, and Working Svo, 

Notes on Elementary Kinematics Svo, boards, 

Graphics of Machine Forces Svo, boards. 

Hedges, K. Modern Lightning Conductors Svo, 

Heermann, P. Dyers' Materials. Trans, by A. C. Wright 12 mo, 

Hellot, Macquer and D' Apligny. Art of Dyeing Wool, Silk and Cotton. Svo, 

Henrici, 0. Skeleton Structures Svo, 

Hering, D. W. Essentials of Physics for College Students Svo, 

Hering-Shaw, A. Domestic Sanitation and Plumbing. Two Vols.. .Svo, 

Hering-Shaw, A. Elementary Science Svo, 

Herrmann, G. The Graphical Statics of Mechanism. Trans, by A. P. 

Smith i2mo, 

H-erzfeld, J. Testing of Yarns and Textile Fabrics Svo, 

Hildebrandt, A. Airships, Past and Present Svo, 

Hildenbrand, B, W. Cable-Making. (Science Series No. 32.) . . . .i6mo, 

Hilditch, T. P. A Concise History of Chemistry izmo, 

Hill, J. W. The Purification of Public Water Supplies. New Edition. 

{hi rrrss.) 

■ Interpretation of Water Analysis U" Press. ) 

Hill, M. J. M. The Theory of Proportion Svo, *2 50 

Hiroi, I. Plate Girder Construction. (Science Series No. 95.1 i6mo, o 50 

Statically-Indeterminate Stresses i2mo, 2 00 

Hirshfeld, C. F. Engineering Thermodynamics. ( Science Series No. 45. ) 

i6mo, o 50 
Hobart, H. M. Heavy Electrical Engineering, Svo, *4 50 

Design of Static Transformers ■.2mo, *2 00 

Electricity -^^^^ -' 00 

Electric Trains «vo, 2 50 



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14 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

Hobart, H. M. Electric Propulsion of Ships 8vo, *2 oo 

Hobart, J. F. Hard Soldering, Soft Soldering and Brazing i2nio, *i oo 

flobbs, W. R. P. The Arithmetic of Electrical Measurements. . . .i2mo, o 50 

Hoff, J. N. Paint and Varnish Facts and Formulas lamo, *i 50 

Hole, W. The Distribution of Gas 8vo, *j 50 

Holley, A. L. Railway Practice folio' 12 00 

Holmes, A. B. The Electric Light Popularly Explained. . .lamo, paper, o 50 

Hopkins, N. M. Experimental Electrochemistry 8vo, *3 00 

Model Engines and Small Boats i2mo, i 25 

Hopkinson, J., Shoolbred, J. ly., and Day, R. E. Dynamic Electricity. 

(Science Series No. 71.) i6mo, o 50 

Horner, J. Metal Turning i2mo, i 50 

Practical Ironf ounding 8vo, *2 00 

Plating and Boiler Making Svo, 3 00 

— ^ Gear Cutting, in Theory and Practice Svo, *3 co 

Houghton, C. E. The Elements of Mechanics of Materials i2mo, *2 co 

Houllevigue, L. The Evolution of the Sciences Svo, *2 00 

Houstoun, R. A, Studies in Light Production i2mo, 2 00 

Hovenden, F. Practical Mathematics for Young Engineers i2mo, *i 00 

Hov.^e, G. Mathematics for the Practical Man i2mo, *i 25 

Howorth, J. Repairing and Riveting Glass, China and Earthenware. 

Svo, paper, *o 50 

Hubbard, E. The Utilization of Wood-v/aste Svo, *2 50 

Htibner, J. Bleaching and Dyeing of Vegetable and Fibrous Materials. 

(Outlines of Industrial Chemistry.) Svo, *5 00 

Hudson, 0. F. Iron and Steel. (Outlines of Industrial Chemistry.). Svo, *2 00 

Humper, W. Calculation of Strains in Girders i2mo, 2 50 

Humphrey, J. C. W. Metallography of Strain. (Metallurgy Series.) 

{In Press.) 

Humphreys, A. C. The Business Features of Engineering Practice.. Svo, *i 25 

Hunter, A. Bridge Work Svo. (In Press.) 

Hurst, G. H. Handbook of the Theory of Color Svo, *2 50 

Dictionary of Chemicals and Raw Products Svo, *3 00 

Lubricating Oils, Fats and Greases Svo, *4 00 

Soaps Svo, *5 00 

Hurst, G. H., and Simmons, W. H. Textile Soaps and Oils Svo, *2 50 

Hurst, H. E., and Lattey, R. T. Text-book of Physics Svo, *3 00 

Also published in three parts. 

Part I. Dynamics and Heat *i 25 

Part II. Sound and Light *i 25 

Part III. Magnetism and Electricity *i 50 

Hutchinson, R. W., Jr. Long Distance Electric Power Transmission. 

i2mo, *3 00 
Hutchinson, R. W., Jr., and Thomas, W. A. Electricity in Mining. i2mo, 

{In Press.) 
Hutchinson, W. B. Patents and How to Make Money Out of Them. 

i2mo, I 25 

Hutton, W, S. Steam-boiler Construction Svo, 6 00 

Practical Engineer's Handbook Svo, 7 00 

The Works' Manager's Handbook Svo, 6 00 






50 


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D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG \ 

Hyde, E. W. Skew Arches. (Science Series No. 15.) i6mo, 

Hyde, F. S. Solvents, Oils, Gums, Waxes 8vo, 

Induction Coils. (Science Series No. 53.) i6mo, 

Ingham, A. E. Gearing. A practical treatise 8vo, 

Ingle, H. Manual of Agricultural Chemistry Svo, 

Inness, C. H. Problems in Machine Design i2mo, 

Air Compressors and Blowing Engines i2mo, 

Centrifugal Pumps i2mo, 

The Fan i2mo, 

Isherwood, B. F. Engineering Precedents for Steam Machinery . . .8vo, 
Ivatts, E. B. Railway Management at Stations 8vo, 

Jacob, A., and Gould, E. S. On the Designing and Construction of 

Storage Reservoirs. (Science Series No. 6) i6mo, o 5 

Jannettaz, E. Guide to the Determination of Rocks. Trans, by G. W. 

Plympton i2mo, 

Jehl, F. Manufacture of Carbons 8vo, 

Jennings, A. S. Commercial Paints and Painting. (Westminster Series.) 

8vo, 

Jennison, F. H. The Manufacture of Lake Pigments Svo, 

Jepson, G. Cams and the Principles of their Construction Svo, 

Mechanical Drawing Svo {In Preparation.) 

Jockin, W. Arithmetic of the Gold and Silversmith i2mo, 

Johnson, J. H. Arc Lamps and Accessory Apparatus. (Installation 

Manuals Series.) i2mo, 

Johnson, T. M. Ship Wiring and Fitting. (Installation Manuals Series.) 

i2mo, 
Johnson, W. H. The Cultivation and Preparation of Para Rubber . . Svo, 

Johnson, W. McA. The Metallurgy of Nickel (//i Preparation.) 

Johnston, J. F. W., and Cameron, C. Elements of Agricultural Chemistry 

and Geology i2mo, 

Joly, J. Radioactivity and Geology i2mo, 

Jones, H. C. Electrical Nature of Matter and Radioactivity i2mo, 

- — New Era in Chemistry lamo, 

Jones, M. W. Testing Raw Materials Used in Paint i2mo. 

Tones, L., and Scard, F. I. Manufacture of Cane Sugar Svo, 

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Joynson, F. H. Designing and Construction of Machine Gearing . Svo, 
Jiiptner, H. F. V. Siderology : The Science of Iron Svo, 

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Keim, A. W. Prevention of Dampness in Buildings Svo, 

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Algebra and Trigonometry, with a Chapter on Vectors 

Special Algebra Edition 

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Kelsey, W. R. Continuous-current Dynamos and Motors Svo, 

Kemble, W. T., and Underbill, C. R. The Periodic Law and the Hydrogen 

Spectrum Svo, paper, *o 50 



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00 


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75 


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60 


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00 


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00 


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00 


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00 


6 


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75 


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25 


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50 



l6 D. VAN NOSTRAXD CO.'S SHORT TITLE CATALOG 

Kemp, J. F. Handbook of Rocks 8vo, *i 50 

Kendall, E. Twelve Figure Cipher Code 4to, *i2 50 

Kennedy, A. B. W., and Thurston, R. H. Kinematics of Machinery. 

(Science Series No. 54.) i6mo, o 50 

Kennedy, A. B. W., Unwin, W. C, and Idell, F. E. Compressed Air. 

(Science Series No. 106.) i6mo, o 50 

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Single Volumes each, 3 00 

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Kennelly, A. E. Electro-dynamic Machinery 8vo, i 50 

Kent, W. Strength of Materials. (Science Series No. 41.) i6mo, o 50 

Kershaw, J. B. C. Fuel, Water and Gas Analysis Svo, *2 50 

Electrometallurgy. (Westminster Series.) Svo, *2 00 

The Electric Furnace in Iron and Steel Production i2mo, *i 50 

Electro-Thermal Methods of Iron and Steel Production. .. .8vo, *3 00 

Kinzbrunner, C. Alternate Current Windings Svo, *i 50 

Continuous Current Armatures , Svo, *i 50 

Testing of Alternating Current Machines Svo, *2 00 

Kirkaldy, W. G. David Kirkaldy's System of Mechanical Testing. .4to, 10 00 

Kirkbride, J. Engraving for Illustration Svo, *i 50 

Kirkwood, J. P. Filtration of River Waters 4to, 7 50 

Kirschke, A. Gas and Oil Engines i2mo, *i 25 

Klein, J. F. Design of a High-speed Steam-engine Svo, *5 00 

Physical Significance of Entropy Svo, *i 50 

Kleinhans, F. B. Boiler Construction Svo, 3 00 

Knight, R.-Adm. A. M. Modem Seamanship Svo, *7 50 

Half morocco *g 00 

Knox, J. Physico-Chemical Calculations i2mo, *i 00 

Fixation of Atmospheric Nitrogen. (Chemical Monographs, 

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Elnox, W. F. Logarithm Tables (In Preparation.) 

Knott, C. G., and Mackay, J. S. Practical Mathematics Svo, 2 00 

Koester, F. Steam-Electric Power Plants 4to, *5 00 

Hydroelectric Developments and Engineering 4to, *5 00 

KoUer, T. The Utilization of Waste Products Svo, *3 50 

Cosmetics Svo, *2 50 

Kremann, R. Application of the Physico-Chemical Theory to Tech- 
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E. Potts Svo, *2 50 

Kretchmar, K. Yam and Warp Sizing Svo, *4 00 

Lallier, E. V. Elementary Manual of the Steam Engine i2mo, *2 00 

Lambert, T. Lead and Its Compounds Svo, *3 50 

Bone Products and Manures Svo, *3 00 

Lamborn, L. L. Cottonseed Products Svo, *3 00 

— — Modern Soaps, Candles, and Glycerin Svo, *7 50 

Lamprecht, R. Recovery Work After Pit Fires. Trans, by C. Salter. Svo, *4 00 

Lancaster, M. Electric Heating, Cooking and Cleaning Svo, =^1 50 



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D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 17 

Lanchester, F. W. Aerial Flight. Two Volumes. 8vo. 

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Aerial Flight. Vol. II. Aerodonetics *6 . 00 

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Larrabee, C. S. Cipher and Secret Letter and Telegraphic Code. i6mo, o 60 

La Rue, B. F. Swing Bridges. (Science Series No. 107.) i6mo, o 50 

Lassar-Cohn. Dr. Modern Scientific Chemistry. Trans, by M. M. 

Pattison Muir i2mo, *2 00 

Latimer, L. H., Field, C. J., and Howell, J. W. Incandescent Electric 

Lighting. (Science Series No. 57.) i6mo, 

Latta, M. N. Handbook of American Gas-Engineering Practice . . . 8vo, 

— — American Producer Gas Practice 4to, 

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Lawson, W. R. British Railways. A Financial and Commercial 

Survey 8 vo, 

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High School Edition *i 25 

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Binns 4to, 

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Lemstrom, S. Electricity in Agriculture and Horticulture Bvo, 

Letts, E. A. Fundamental Problems in Chemistry Bvo, 

Le Van, W. B. Steam-Engine Indicator. (Science Series No. 78.)i6mo, 
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Carbonization of Coal 8vo, 

Lewis, L. P. Railway Signal Engineering 8vo, 

Lieber, B. F. Lieber's Standard Telegraphic Code 8vo, 

Code. German Edition 8vo, 

Spanish Edition 8vo, 

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Engineering Code 8vo, *i2 50 

Livermore, V. P., and Williams, J. How to Become a Competent Motor- 



*7 


50 


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*I 


50 

50 


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10 


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15 


00 


*2 


50 


15 


00 


10 


00 



Liversedge, A. J. Commercial Engineering 

Livingstone, R. Design and Construction of Commutators 
Mechanical Design and Construction of Generators. 

Lobben, P. Machinists' and Draftsmen's Handbook 

Lockwood, T. D. Electricity, Magnetism, and Electro-telegraph 



Svo, 


.S 00 


8vo, 


'2 25 


Bvo, 


■^3 50 


Svo, 


i 50 


Svo, 


2 50 



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Loring, A. E. A Handbook of the Electromagnetic Telegraph .... i6mo o 50 

' Handbook. (Science Series No. 39.) i6mo, o 50 

Low, D. A. Applied Mechanics (Elementary) i6mo, o 80 

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in collaboration with the corps of specialists. 

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Macewen, H. A. Food Inspection 8vo, *2 50 

Mackenzie, N. F. Notes on Irrigation Works 8vo, *2 50 

Mackie, J. How to Make a Woolen Mill Pay 8vo, *2 00 

Mackrow, C. Naval Architect's and Shipbuilder's Pocket-book. 

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No. 10.) i6mo, 

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Marlow, T. G. Drying Machinery and Practice 8vo, *5 00 

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' Reinforced Concrete Compression Member Diagram. Moimted on 

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Marsh, C. F., and Dunn, W. Manual of Reinforced Concrete and Con- 
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8vo, *2 00 

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Martin, W. D. Hints to Engineers i2mo, *i 00 

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McCuUough, R. S. Mechanical Theory of Heat 8vo, 

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Vol. II. Varnish Materials and Oil Varnish Making *4 00 

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i6mo, 

McMechen, F. L. Tests for Ores, Minerals and Metals i2mo, 

McPherson, J. A. Water-works Distribution 8vo, 

Melick, C. W. Dairy Laboratory Guide i2mo, 

Merck, E. Chemical Reagents ; Their Purity and Tests. Trans, by 

H. E. Schenck 8vo, 

Merivale, J. H, Notes and Formulae for Mining Students lamo, 

Merritt, Wm. H. Field Testing for Gold and Silver i6mo, leather, 

Messer, W. A. Railway Permanent Way 8vo (/// Prcs-s.) 

Meyer, J. G. A., and Pecker, C. G. Mechanical Drawing and Machine 

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Michell, S. Mine Drainage 8vo, 

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Robson 8vo, " 

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Milroy, M. E. W. Home Lace-making. . i2mo, 'i 00 

Minifie, W. Mechanical Drawing 8vo, *4 00 

Mitchell, C. A. Mineral and Aerated Waters. . 8vo, *3 00 



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Mitchell, C. A., and Prideaux, R. M. Fibres Used in Textile and Allied 

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Advanced Course *2 50 

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Moore, E. C. S. New Tables for the Complete Solution of Ganguillet and 

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Morecroft, J. H., and Hehre, F. W. Short Course in Electrical Testing. 

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Morgan, A. P. Wireless Telegraph Apparatus for Amateurs i2mo, *i 50 

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Moss, S. A. Elements of Gas Engine Design. Science Series No. 121. i6mo, o 50 

TheLay-out of Corliss Valve Gears. Science Series No. 119. i6mo, o 50 

Mulford, A. C. Boundaries and Landmarks i2mo, *i 00 

Mullin, J. P. Modem Moulding and Pattern-making i2mo, 2 50 

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Recent Cotton Mill Construction i2mo, 2 00 

Neave, G. B., and Heilbron, L M. Identification of Organic Compounds. 

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Newall, J. W. Drawing, Sizing and Cutting Bevel-gears 8vo, i 50 

Nicol, G. Ship Construction and Calculations 8vo, *4 50 

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Noll, A. How to Wire Buildings i2mo, i 50 

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Petrol Air Gas i^mo, *o 75 



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Ohm, G. S., and Lockwood, T. D. Galvanic Circuit. Translated by 

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Olsson, A. Motor Control, in Turret Turning and Gun Elevating. (U. S. 

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Ormsby, M. T. M. Surveying i2mo, i 50 

Oudin, M. A. Standard Polyphase Apparatus and Systems 8vo, *3 00 

Owen, D. Recent Physical Research 8vo, ''i 50 

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Pamely, C. Colliery Manager's Handbook 8vo, *io 00 

Parker, P. A. M. The Control of Water 8vo, ^^5 00 

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22 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

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Rathbone, R. L. B. Simple Jewellery 8vo, 

Rateau, A. Flow of Steam through Nozzles and Orifices. Trans, by H. 

B. Brydon 8vo 

Rausenberger, F. The Theory of the Recoil of Guns 8vo, 

Rautenstrauch, W. Notes on the Elements of Machine Design. 8 vo, boards, 
Rautenstrauch, W., and Williams, J. T. Machine Drafting and Empirical 

Design. 

Part I. Machine Drafting 8vo, *i 25 

Part II. Empirical Design {In Preparation.) 

Raymond, E. B. Alternating Current Engineering i2mo, 

Rayner, H. Silk Throwing and Waste Silk Spinning 8vo, 

Recipes for the Color, Paint, Varnish, Oil, Soap and Drysaltery Trades . 8vo, 

Recipes for Flint Glass Making i2mo, 

Redfern, J. B., and Savin, J. Bells, Telephones (Installation Manuals 

Series.) i6mo, 

Redgrove, H. S. Experimental Mensuration lamo, 

Redwood, B. Petroleum. (Science Series No. 92.) i6mo, 

Reed, S. Turbines Applied to Marine Propulsion *5 

Reed's Engineers' Handbook 8vo, 

Key to the Nineteenth Edition of Reed's Engineers' Handbook. .8vo, 

Useful Hints to Sea-going Engineers i2rao, 

Marine Boilers i2mo, 

Guide to the Use of the Slide Valve i2mo, 

Reinhardt, C. W. Lettering for Draftsmen, Engineers, and Students. 

oblong 4to, boards, 

■ The Technic of Mechanical Drafting oblong 4to, boards, 

Reiser, F. Hardening and Tempering of Steel. Trans, by A. Morris and 

H. Robson i2mo, *2 50 

Reiser, N. Faults in the Manufacture of Woolen Goods. Trans, by A. 

Morris and H. Robson ,8vo, *2 50 

Spinning and Weaving Calculations 8vo, *5 00 

Renwick, W. G. Marble and Marble Working 8vo, 5 00 



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*i 


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25 





50 


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24 D. \'AX XOSTRAXD CO.'S SHORT TITLE CATALOG 

Reynolds, 0., and Idell, F. E. Triple Expansion Engines. (Science 

Series No. 99.) i6mo, o 50 

Rhead, G. F. Simple Structural Woodwork i2mo, *i 00 

Rhodes, H. J. Art of Lithography 8vo, 3 50 

Rice, J. M., and Johnson, W. W. A New Method of Obtaining the Differ- 
ential of Functions i2mo, o 50 

Richards, W. A. Forging of Iron and Steel (/jz Press.) 

Richards, W. A., and North, H. B. Manual of Cement Testing. . . . i2mo, 

Richardson, J. The Modern Steam Engine 8vo, 

Richardson, S. S. Magnetism and Electricity i2mo, 

Rideal, S. Glue and Glue Testing 8vo, 

Rimmer, E. J. Boiler Explosions, Collapses and Mishaps 8vo, 

Rings, F. Concrete in Theory and Practice i2mo, 

Reinforced Concrete Bridges 4to, 

Ripper, W. Course of Instruction in Machine Drawing folio, 

Roberts, F. C. Figure of the Earth. (Science Series No. 79.) i6mo, 

Roberts, J., Jr. Laboratory Work in Electrical Engineering Bvo, 

Robertson, L. S. Water-tube Boilers Bvo, 

Robinson, J. B. Architectural Composition Bvo, 

Robinson, S. W. Practical Treatise on the Teeth of Wheels. (Science 

Series No. 24.) i6mo, 

— ^ — Railroad Economics. (Science Series No. 59.) i6mo, 

' — ■ — Wrought Iron Bridge Members. (Science Series No. 60.) i6mo, 

Robson, J. H. Machine Drawing and Sketching 8vo, 

Roebling, J. A. Long and Short Span Railway Bridges folio, 

Rogers, A. A Laboratory Guide of Industrial Chemistry i2mo, 

Rogers, A., and Aubert, A. B. Industrial Chemistry Bvo, 

Rogers, F. Magnetism of Iron Vessels. (Science Series No. 30.) . i6mo, 
Rohland, P. Colloidal and Crystalloidal State of Matter. Trans, by 

W. J. Britland and H. E. Potts i2mo, 

Rollinsj W. Notes on X-Light Bvo, 

Rollinson, C. Alphabets Oblong, i2mo, 

Rose, J. The Pattern-makers' Assistant Bvo, 

Key to Engines and Engine-running i2mo. 

Rose, T. K. The Precious Metals. (Westminster Series.) Bvo, 

Rosenhain, W. Glass Manufacture. (Westminster Series.) Bvo, *2 00 

Physical Metallurgy, An Introduction to. (Metallurgy Series.) 

8vo, (hi Press.) 

Ross, W. A. Blowpipe in Chemistry and Metallurgy i2mo, 

Roth. Physical Chemistry Bvo, 

Rouillion, L. The Economics of Manual Training Bvo, 

Rowan, F. J. Practical Physics of the Modern Steam-boiler Bvo, 

and Idell, F. E. Boiler Incrustation and Corrosion. (Science 

Series No. 27,) i6mo, 

Roxburgh, W. General Foundry Practice. (Westminster Series.) .Bvo, 
Ruhmer, E. Wireless Telephony. Trans, by J. Erskine-Murray . .Bvo, 
Russell, A. Theory of Electric Cables and Networks Bvo, 

Sabine, R. History and Progress of the Electric Telegraph i2mo, 

Saeltzer, A. Treatise on Acoustics 12^0, 



*I 


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D. VAX NOSTRAND CO.'S SHORT TITLE CATALOG 25 

Sanford, P. G. Nitro-explosives Svo, 

Saunders, C. H. Handbook of Practical Mechanics i6mo, 

leather, 

Saunnier, C. Watchmaker's Handbook i2mo, 

Sayers, H. M. Brakes for Tram Cars Svo, 

Scheele, C. W. Chemical Essays Svo, 

Scheithauer, W. Shale Oils and Tars Svo, 

Schellen, H. Magneto-electric and Dynamo-electric Machines .... Svo, 

Scherer, R. Casein. Trans, by C. Salter Svo, 

Schidrowitz, P. Rubber, Its Production and Industrial Uses Svo, 

Schindler, K. Iron and Steel Construction Works i2mo, 

Schmall, C. N. First Coiurse in Analytic Geometry, Plane and Solid. 

i2mo, half leather, 
Schmall, C. N., and Shack, S. M. Elements of Plane Geometry. . . i2mo, 

Schmeer, L. Flow of Water Svo, 

Schumann, F. A Manual of Heating and Ventilation. . . .lamo, leather, 

Schwarz, E. H. L. Causal Geology Svo, 

Schweizer, V. Distillation of Resins Svo, 

Scott, W. W. Qualitative Analysis. A Laboratory Manual Svo, 

Scribner, J. M. Engineers' and Mechanics' Companion. .i6mo, leather, 
Scudder, H^ Electrical Conductivity and Ionization Constants of 

Organic Compounds Svo, 

Searle, A. B. Modern Brickraaking Svo, 

Cement, Concrete and Bricks Svo, 

Searle, G. M. "Sumners' Method." Condensed and Improved. 

(Science Series No. 124.) i6mo, 

Seaton, A. E. Manual of Marine Engineering Svo 

Seaton, A. E., and Rounthwaite, H. M. Pocket-book of Marine Engi- 
neering i6mo, leather, 3 50 

Seeligmann, T., Torrilhon, G. L., and Falconnet, H. India Rubber and 

Gutta Percha. Trans, by J. G. Mcintosh Svo, 

Seidell, A. Solubilities of Inorganic and Organic Substances Svo, 

Seligman, R. Aluminum. (Metallurgy Series.) (/;/ Press.) 

Sellew, W. H. Steel Rails 4to, 

Senter, G. Outlines of Physical Chemistry i2mo, 

Text-book of Inorganic Chemistry i2mo, 

Sever, G. F. Electric Engineering Experiments Svo, boards. 

Sever, G. F., and Townsend, F. Laboratory and Factory Tests in Elec- 
trical Engineering Svo, 

Sewall, C. H. Wireless Telegraphy Svo, 

Lessons in Telegraphy i2mo, 

Sewell, T. Elements of Electrical Engineering Svo, 

The Construction of Dynamos. Svo, 

Sexton, A. H. Fuel and Refractory Materials i2mo, 

Chemistry of the Materials of Engineering i2mo, 

Alloys (Non-Ferrous) Svo, 

The Metallurgy of Iron and Steel Svo, 

Seymour, A. Practical Lithography Svo, 

Modern Printing Inks Svo, 



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50 


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50 



26 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

Shaw, Henry S. H. Mechanical Integrators. (Science Series No. 83.) 

i6mo, 

Shaw, S. History of the Staffordshire Potteries 8vo, 

Chemistry of Compounds Used in Porcelain Manufacture. .. .Svo, 

Shaw, W. N. Forecasting Weather Svo, 

Sheldon, S., and Hausmann, E. Direct Current Machines i2mo, *2 50 

Alternating Current Machines i2mo, *2 50 

Sheldon, S., and Hausmann, E. Electric Traction and Transmission 

Engineering i2mo, *2 50 

Sheriff, F. F. Oil Merchants' Manual lamo, 

Shields, J. E. Notes on Engineering Construction lamo, 

Shreve, S. H. Strength of Bridges and Roofs Svo, 

Shunk, W. F. The Field Engineer i2mo, morocco, 

Simmons, W. H., and Appleton, H. A. Handbook of Soap Manufacture, 

Svo, 

Simmons, W. H., and Mitchell, C. A. Edible Fats and Oils Svo, 

Simms, F. W. The Principles and Practice of Levelling Svo, 

Practical Tunneling Svo, 

Simpson, G. The Naval Constructor i2mo, morocco, 

Simpson, W. Foundations Svo. (/« Press.) 

Sinclair, A. Development of the Locomotive Engine. . .8vo, half leather, 
Twentieth Century Locomotive Svo, half leather, 

Sindall, R. W., and Bacon, W. N. The Testing of Wood Pulp Svo, 

Sindall, R. W. Manufacture of Paper. (Westminster Series.). .. .Svo, 

Sloane, T. O'C. Elementary Electrical Calculations i2mo, 

Smallwood, J. C. Mechanical Laboratory Methods. .. .i2mo, leather, 

Smith, C. A. M. Handbook of Testing, MATERIALS Svo, 

Smith, C. A. M., and Warren, A. G. New Steam Tables Svo, 

Smith, C. F. Practical Alternating Currents and Testing Svo, 

Practical Testing of Dynamos and Motors Svo, 

Smith, F. E. Handbook of General Instruction for Mechanics . . . i2mo. 
Smith, H. G. Minerals and the Microscope 

Smith, J. C. Manufacture of Paint Svo, *3 oc 

Paint and Painting Defects 

Smith, R. H. Principles of Machine Work i2mo, *3 00 

Elements of Machine Work i2mo, *2 00 

Smith, W. Chemistry of Hat Manufacturing i2mo, *3 00 

Snell, A. T. Electric Motive Power Svo, *4 00 

Snow, W. G. Pocketbook of Steam Heating and Ventilation. {In Press.) 
Snow, W. G., and Nolan, T. Ventilation of Buildings. (Science Series 

No. 5.) ; i6mo, o 50 

Soddy, F. Radioactivity Svo, 

Solomon, M. Electric Lamps. (Westminster Series.) Svo, 

Somerscales, A. N. Mechanics for Marine Engineers i2mo, 

Mechanical and Marine Engineering Science Svo, 

Sothern, J. W. The Marine Steam Turbine Svo, 

• Verbal Notes and Sketches for Marine Engineers Svo, 

Sothern, J. W., and Sothern, R. M. Elementary Mathematics for 

Marine Engineers i2mo, 

■ Simple Problems in Marine Engineering Design i2mo, 



*3 


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7 


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00 


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D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 27 

Southcombe, J. E. Chemistry of the Oil Industries. (Outlines of In- 
dustrial Chemistry.) 8vo, *3 oc> 

Soxhlet, D. H. Dyeing and Staining Marble. Trans, by A. Morris and 

H. Robson 8vo, *2 50 

Spang, H. W. A Practical Treatise on Lightning Protection i2mo, i 00 

Spangenburg, L. Fatigue of Metals. Translated by S. H. Shreve. 

(Science Series No. 23.) i6mo, o 50 

Specht, G. J., Hardy, A. S., McMaster, J. B., and Walling. Topographical 

Surveying. (Science Series No. 72.) i6mo, o 50 

Speyers, C. L. Text-book of Physical Chemistry 8vo, *2 25 

Sprague, E. H. Hydraulics lamo, i 25 

Stahl, A. W. Transmission of Power. (Science Series No. 28.) . i6mo, 

Stahl, A. W., and Woods, A. T. Elementary Mechanism i2mo, *2 00 

Staley, C, and Pierson, G. S. The Separate System of Sewerage. . .8vo, *3 00 

Standage, H. C. Leatherworkers' Manual 8vo, *3 50 

Sealing Waxes, Wafers, and Other Adhesives 8vo, *2 00 

Agglutinants of all Kinds for all Purposes i2mo, *3 50 

Stanley, H. Practical Applied Physics {In Press.) 

Stansbie, J. H. Iron and Steel. (Westminster Series.) Bvo, *2 00 

Steadman, F. M. Unit Photography and Actinometry (In Press.) 

Stecher, G. E. Cork, Its Origin and Industrial Uses i2mo, i 00 

Steinman, D. B. Suspension Bridges and Cantilevers. (Science Series 

No. 127.) o 50 

Stevens, H. P. Paper Mill Chemist i6mo, *2 50 

Stevens, J. S. Precision of Measurements (/// Press.) 

Stevenson, J. L. Blast-Furnace Calculations i2mo, leather, *2 00 

Stewart, A. Modern Polyphase Machinery i2mo, *2 00 

Stewart, G. Modem Steam Traps ; i2mo, *i 25 

Stiles, A. Tables for Field Engineers i2mo, i 00 

Stillman, P. Steam-engine Indicator i2mo, i oo 

Stodola, A, Steam Turbines. Trans, by L. C. Loewenstein 8vo, *5 00 

Stone, H. The Timbers of Commerce 8vo, 3 50 

Stone, Gen. R. New Roads and Road Laws i2mo, i 00 

Stopes, M. Ancient Plants 8vo, *2 00 

The Study of Plant Life 8vo, *2 00 

Stumpf, Prof. Una-Flow of Steam Engine 4to, *3 50 

Sudborough, J. J., and James, T. C. Practical Organic Chemistry. . i2mo, *2 00 

Suffling, E. R. Treatise on the Art of Glass Painting 8vo, *3 50 

Swan, K. Patents, Designs and Trade Marks. (Westminster Series.). 

8vo, *2 00 
Swinburne, J., Wordingham, C. H., and Martin, T. C. Electric Currents. 

(Science Series No. 109.) i6mo, o 50 

Swoope, C. W. Lessons in Practical Electricity i2mo, *2 00 

Tailf er, L. Bleaching Linen and Cotton Yarn and Fabrics 8vo, *$ 00 

Tate, J. S. Surcharged and Different Forms of Retaining-walls. (Science 

Series No. 7.) i6mo, o 50 

Taylor, E. N. Small Water Supplies i2mo, *2 00 

Templeton, W. Practical Mechanic's Workshop Companion. 

i2mo, morocco, 2 00 



28 D. VAN NOSTKAND CO.'S SHORT TITLE CATALOG 

Tenney, E. H. Test Methods for Steam Power Plants (/;; Press.) 

Terry, H. L. India Rubber and its Manufacture. (Westminster Series.) 

8vo, 
Thayer, H. R. Structural Design. 8vo. 

Vol. I. Elements of Structural Design 

Vol. IL Design of Simple Structures {In Preparation.) 

Vol. in. Design of Advanced Structures {In Preparation.) 

Thiess, J. B., and Joy, G. A. Toll Telephone Practice 8vo, 

Thorn, C, and Jones, W. H. Telegraphic Connections.. . oblong, i2mo, 

Thomas, C. W. Paper-makers' Handbook {In Press.) 

Thompson, A. B. Oil Fields of Russia 4to, 

Petroleum Mining and Oil Field Development Svo, 

Thompson, S. P. Dynamo Electric Machines. (Science Series No. 75.) 

i6mo, 

Thompson, W. P. Handbook of Patent Law of All Countries i6mo, 

Thomson, G. S. Milk and Cream Testing i2mo, 

Modern Sanitary Engineering, House Drainage, etc Svo, 

Thornley, T. Cotton Combing Machines Svo, 

Cotton Waste Svo, 

Cotton Spinning. Svo. 

First Year 

Second Year 

Third Year *2 

Thurso, J. W. Modern Turbine Practice Svo, 

Tidy, C. Meymott. Treatment of Sewage. (Science Series No. 94.) i6mo, 
Tillmans, J. Water Purification and Sewage Disposal. Trans, by 
Hugh S. Taylor Svo, 

Tinney, W. H. Gold-mining Machinery Svo, 

Titherley, A. W. Laboratory Course of Organic Chemistry Svo, 

Toch, M. Chemistry and Technology of Mixed Paints Svo, 

Materials for Permanent Painting 1 2mo, 

Chemistry and Technology of Mixed Paints. (In two volumes.) 

{In Press.) 
Tod, J., and McGibbon, W. C. Marine Engineers' Board of Trade 

Examinations Svo, 

Todd, J., and Whall, W. B. Practical Seamanship Svo, 

Tonge, J. Coal. (Westminster Series.) Svo, 

Townsend, F. Alternating Current Engineering Svo, boards, 

Townsend, J. Ionization of Gases by Collision Svo, 

Transactions of the American Institute of Chemical Engineers, Svo. 

Vol. I. 1908 

Vol. IL 1909 

Vol. m. 1910 

Vol. IV. 1911 

Vol. V. 1912 

Vol. VI. 1913 *6 00 

Traverse Tables. (Science Series No. 115.) i6mo, o 50 

morocco, i 00 
Treiber, E. Foundry Machinery. Trans, by C. Salter xamo, i as 



^3 


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00 


*6 


00 


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00 



D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 29 

Trinks, W., and Housum, C. Shaft Governors. (Science Series No. 122.) 

i6mo, 50 

Trowbridge, W. P. Turbine Wheels. (Science Series No. 44.) . . i6mo, o 50 

Tucker, J. H. A Manual of Sugar Analysis 8vo, 3 50 

Tunner, P. A. Treatise on Roll-turning. Trans, by J. B. Pearse. 

8vo, text and folio atlas, 10 00 
Turnbull, Jr., J., and Robinson, S. W. A Treatise on the Compound 

Steam-engine. (Science Series No. 8.) i6mo, 

Turrill, S. M. Elementary Course in Perspective i2mo, *i 25 

Underbill, C. R. Solenoids, Electromagnets and Electromagnetic Wind- 
ings i2mo, *2 00 

Underwood, N., and Sullivan, T. V. Chemistry and Technology of 

Printing Inks (In Press.) 

Urquhart, J. W. Electric Light Fitting i2mo, 2 00 

— ■ — Electro-plating i2mo, 2 00 

— — Electrotyping i2n:o, 2 00 

Electric Ship Lighting i2mo, 3 00 

Usborne, P. O. G. Design of Simple Steel Bridges 8vo, *4 00 

Vacher, F. Food Inspector's Handbook 

Van Nostrand's Chemical AnnuaL Third issue 1913. .. .leather, i2mo, *2 50 

Year Book of Mechanical Engineering Data (In Press.) 

Van Wagenen, T. F. Manual of Hydraulic Mining i6mo, i 00 

Vega, Baron Von. Logarithmic Tables Svo, cloth, 2 00 

half morroco, 2 50 

Vincent, C. Ammonia and its Compounds. Trans, by M. J. Salter . Svo, *2 00 

Volk, C. Haulage and Winding Appliances Svo, *4 00 

Von Georgievics, G. Chemical Technology of Textile Fibres. Trans. 

by C. Salter Svo, *4 5o 

Chemistry of Dyestuffs. Trans, by C. Salter Svo, *4 50 

V«se, G. L. Graphic Method for Solving Certain Questions in Arithmetic 

and Algebra (Science Series No. 16.) i6mo, o 50 

Vosmaer, A. Ozone (In Press.) 

Wabner, R. Ventilation in Mines. Trans, by C. Salter Svo, *4 50 

Wade, E. J. Secondary Batteries Svo, *4 00 

Wadmore, T. M. Elementary Chemical Theory i2mo, *i 50 

Wadsworth, C. Primary Battery Ignition i2mo, *o 50 

Wagner, E. Preserving Fruits, Vegetables, and Meat i2mo, *2 50 

Waldram, P. J. Principles of Structural Mechanics i2mo, *3 00 

Walker, F. Aerial Navigation Svo, 2 00 

Dynamo Building. (Science Series No. q8.) i6mo, o ^o 

Walker, F. Electric Lighting for Marine Engineers Svo, 2 00 

Walker, J. Organic Chemistry for Students of Medicine Svo, *2 50 

Walker, S. F. Steam Boilers, Engines and Turbines Svo, 3 00 

Refrigeration, Heating and Ventilation on Shipboard i2mo, *2 00 

Electricity in Mining 8vo, *3 50 

Wallis-Tayler, A. J. Bearings and Lubrication Svo, *i 50 

Aerial or Wire Ropeways Svo, *3 00 



30 D. VAN NOSTRAND CO.'S SHORT TITLE CATALOG 

Wallis-Tayler, A.J. Pocket Book of Refrigeration and Ice Making. i2mo, 

Motor Cars 8vo, 

Motor Vehicles for Business Purposes 8vo, 

Refrigeration, Cold Storage and Ice-Making 8vo, 

Sugar Machinery i2mo, 

Wanklyn, J. A. Water Analysis. i2mo, 

Wansbrough, W. D. The A B C of the Differential Calculus lamo, 

Slide Valves lamo, 

Waring, Jr., G. E. Sanitary Conditions. (Science Series No. 31.) . i6mo, 

Sewerage and Land Drainage *6 00 

Waring, Jr., G. E. Modern Methods of Sewage Disposal lamo, 

How to Drain a House lamo, 

Warnes, A. R. Coal Tar Distillation Bvo, 

Warren, F. D. Handbook on Reinforced Concrete lamo, 

Watkins, A. Photography. (Westminster Series.) Bvo, 

Watson, E. P. Small Engines and Boilers i2mo, 

Watt, A. Electro-plating and Electro-refining of Metals 8va, 

Electro-metallurgy i2mo, 

The Art of Soap Making 8vo, 

Leather Manufacture 8vo, 

Paper-Making 8vo, 

Weale, J. Dictionary of Terms Used in Architecture •. . . i2mo, 

Weale's Scientific and Technical Series. (Complete list sent on appli- 
cation.) 
Weather and Weather Instruments i2mo, 

paper, 
Webb, H. L. Guide to the Testing of Insulated Wires and Cables. i2mo, 

Webber, W. H. Y. Town Gas. (Westminster Series.) Bvo, 

Weisbach, J. A Manual of Theoretical Mechanics Bvo, 

sheep, 
Weisbach, J., and Herrmann, G. Mechanics of Air Machinery .... Bvo, 

Welch, W. Correct Lettering {In Press.) 

Weston, E. B. Loss of Head Due to Friction of Water in Pipes. .i2mo, 

Weymouth, F. M. Drum Armatures and Commutators Bvo, 

Wheatley, 0. Ornamental Cement Work (In Press.) 

Wheeler, J. B. Art of War i2mo, 

Field Fortifications i2mo, 

Whipple, S. An Elementary and Practical Treatise on Bridge Building. 

Bvo, 

White, A. T. Toothed Gearing i2mo, 

White, C. H. Methods of Metallurgical Analysis (In Press.) 

Whithard, P. Illuminating and Missal Painting i2mo, 

Wilcox, R. M. Cantilever Bridges. (Science Series No. 25.) . . . .i6mo, 

Wilda, H. Steam Turbines. Trans, by C. Salter i2mo, 

Cranes and Hoists. Trans, by C. Salter i2mo, 

Wilkinson, H. D. Submarine Cable Laying and Repairing Bvo, 

Williamson, J., and Blackadder, H. Survejang Bvo, (/;/ Press.) 

Williamson, R. S. On the Use of the Barometer 4to, 

Practical Tables in Meteorology and Hypsometery 4to, 

Wilson, F. J., and Heilbron, I. M. Chemical Theory and Calculations. 



I 


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*2 


50 


-2 


50 


*2 


00 


I 


25 


*4 


50 


I 


00 


3 


00 


*4 


00 


3 


00 


2 


50 


I 


00 





50 


I 


00 


*2 


00 


*6 


00 


*7 


50 


*3 


75 


*i 


50 


*3 


00 


I 


75 


I 


75 


3 


00 


*i 


25 


I 


50 





50 


I 


25 


I 


25 


*6 


00 


15 


00 


2 


50 



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